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Theorem itunitc 9435
 Description: The union of all union iterates creates the transitive closure; compare trcl 8777. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Hypothesis
Ref Expression
ituni.u 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
Assertion
Ref Expression
itunitc (TC‘𝐴) = ran (𝑈𝐴)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem itunitc
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6352 . . . 4 (𝑎 = 𝐴 → (TC‘𝑎) = (TC‘𝐴))
2 fveq2 6352 . . . . . 6 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
32rneqd 5508 . . . . 5 (𝑎 = 𝐴 → ran (𝑈𝑎) = ran (𝑈𝐴))
43unieqd 4598 . . . 4 (𝑎 = 𝐴 ran (𝑈𝑎) = ran (𝑈𝐴))
51, 4eqeq12d 2775 . . 3 (𝑎 = 𝐴 → ((TC‘𝑎) = ran (𝑈𝑎) ↔ (TC‘𝐴) = ran (𝑈𝐴)))
6 vex 3343 . . . . . . 7 𝑎 ∈ V
7 ituni.u . . . . . . . 8 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
87ituni0 9432 . . . . . . 7 (𝑎 ∈ V → ((𝑈𝑎)‘∅) = 𝑎)
96, 8ax-mp 5 . . . . . 6 ((𝑈𝑎)‘∅) = 𝑎
10 fvssunirn 6378 . . . . . 6 ((𝑈𝑎)‘∅) ⊆ ran (𝑈𝑎)
119, 10eqsstr3i 3777 . . . . 5 𝑎 ran (𝑈𝑎)
12 dftr3 4908 . . . . . 6 (Tr ran (𝑈𝑎) ↔ ∀𝑏 ran (𝑈𝑎)𝑏 ran (𝑈𝑎))
137itunifn 9431 . . . . . . . 8 (𝑎 ∈ V → (𝑈𝑎) Fn ω)
14 fnunirn 6674 . . . . . . . 8 ((𝑈𝑎) Fn ω → (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐)))
156, 13, 14mp2b 10 . . . . . . 7 (𝑏 ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐))
16 elssuni 4619 . . . . . . . . 9 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ((𝑈𝑎)‘𝑐))
177itunisuc 9433 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) = ((𝑈𝑎)‘𝑐)
18 fvssunirn 6378 . . . . . . . . . 10 ((𝑈𝑎)‘suc 𝑐) ⊆ ran (𝑈𝑎)
1917, 18eqsstr3i 3777 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ ran (𝑈𝑎)
2016, 19syl6ss 3756 . . . . . . . 8 (𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2120rexlimivw 3167 . . . . . . 7 (∃𝑐 ∈ ω 𝑏 ∈ ((𝑈𝑎)‘𝑐) → 𝑏 ran (𝑈𝑎))
2215, 21sylbi 207 . . . . . 6 (𝑏 ran (𝑈𝑎) → 𝑏 ran (𝑈𝑎))
2312, 22mprgbir 3065 . . . . 5 Tr ran (𝑈𝑎)
24 tcmin 8790 . . . . . 6 (𝑎 ∈ V → ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎)))
256, 24ax-mp 5 . . . . 5 ((𝑎 ran (𝑈𝑎) ∧ Tr ran (𝑈𝑎)) → (TC‘𝑎) ⊆ ran (𝑈𝑎))
2611, 23, 25mp2an 710 . . . 4 (TC‘𝑎) ⊆ ran (𝑈𝑎)
27 unissb 4621 . . . . 5 ( ran (𝑈𝑎) ⊆ (TC‘𝑎) ↔ ∀𝑏 ∈ ran (𝑈𝑎)𝑏 ⊆ (TC‘𝑎))
28 fvelrnb 6405 . . . . . . 7 ((𝑈𝑎) Fn ω → (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏))
296, 13, 28mp2b 10 . . . . . 6 (𝑏 ∈ ran (𝑈𝑎) ↔ ∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏)
307itunitc1 9434 . . . . . . . . 9 ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎)
3130a1i 11 . . . . . . . 8 (𝑐 ∈ ω → ((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎))
32 sseq1 3767 . . . . . . . 8 (((𝑈𝑎)‘𝑐) = 𝑏 → (((𝑈𝑎)‘𝑐) ⊆ (TC‘𝑎) ↔ 𝑏 ⊆ (TC‘𝑎)))
3331, 32syl5ibcom 235 . . . . . . 7 (𝑐 ∈ ω → (((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎)))
3433rexlimiv 3165 . . . . . 6 (∃𝑐 ∈ ω ((𝑈𝑎)‘𝑐) = 𝑏𝑏 ⊆ (TC‘𝑎))
3529, 34sylbi 207 . . . . 5 (𝑏 ∈ ran (𝑈𝑎) → 𝑏 ⊆ (TC‘𝑎))
3627, 35mprgbir 3065 . . . 4 ran (𝑈𝑎) ⊆ (TC‘𝑎)
3726, 36eqssi 3760 . . 3 (TC‘𝑎) = ran (𝑈𝑎)
385, 37vtoclg 3406 . 2 (𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
39 rn0 5532 . . . . 5 ran ∅ = ∅
4039unieqi 4597 . . . 4 ran ∅ =
41 uni0 4617 . . . 4 ∅ = ∅
4240, 41eqtr2i 2783 . . 3 ∅ = ran ∅
43 fvprc 6346 . . 3 𝐴 ∈ V → (TC‘𝐴) = ∅)
44 fvprc 6346 . . . . 5 𝐴 ∈ V → (𝑈𝐴) = ∅)
4544rneqd 5508 . . . 4 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4645unieqd 4598 . . 3 𝐴 ∈ V → ran (𝑈𝐴) = ran ∅)
4742, 43, 463eqtr4a 2820 . 2 𝐴 ∈ V → (TC‘𝐴) = ran (𝑈𝐴))
4838, 47pm2.61i 176 1 (TC‘𝐴) = ran (𝑈𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  ∅c0 4058  ∪ cuni 4588   ↦ cmpt 4881  Tr wtr 4904  ran crn 5267   ↾ cres 5268  suc csuc 5886   Fn wfn 6044  ‘cfv 6049  ωcom 7230  reccrdg 7674  TCctc 8785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-tc 8786 This theorem is referenced by:  hsmexlem5  9444
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