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Theorem upgrspanop 26234
 Description: A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.)
Hypotheses
Ref Expression
uhgrspanop.v 𝑉 = (Vtx‘𝐺)
uhgrspanop.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrspanop (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)

Proof of Theorem upgrspanop
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 uhgrspanop.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uhgrspanop.e . . . . 5 𝐸 = (iEdg‘𝐺)
3 vex 3234 . . . . . 6 𝑔 ∈ V
43a1i 11 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝑔 ∈ V)
5 simprl 809 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → (Vtx‘𝑔) = 𝑉)
6 simprr 811 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → (iEdg‘𝑔) = (𝐸𝐴))
7 simpl 472 . . . . 5 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝐺 ∈ UPGraph)
81, 2, 4, 5, 6, 7upgrspan 26230 . . . 4 ((𝐺 ∈ UPGraph ∧ ((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴))) → 𝑔 ∈ UPGraph)
98ex 449 . . 3 (𝐺 ∈ UPGraph → (((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
109alrimiv 1895 . 2 (𝐺 ∈ UPGraph → ∀𝑔(((Vtx‘𝑔) = 𝑉 ∧ (iEdg‘𝑔) = (𝐸𝐴)) → 𝑔 ∈ UPGraph))
11 fvex 6239 . . . 4 (Vtx‘𝐺) ∈ V
121, 11eqeltri 2726 . . 3 𝑉 ∈ V
1312a1i 11 . 2 (𝐺 ∈ UPGraph → 𝑉 ∈ V)
14 fvex 6239 . . . . 5 (iEdg‘𝐺) ∈ V
152, 14eqeltri 2726 . . . 4 𝐸 ∈ V
1615resex 5478 . . 3 (𝐸𝐴) ∈ V
1716a1i 11 . 2 (𝐺 ∈ UPGraph → (𝐸𝐴) ∈ V)
1810, 13, 17gropeld 25970 1 (𝐺 ∈ UPGraph → ⟨𝑉, (𝐸𝐴)⟩ ∈ UPGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   ↾ cres 5145  ‘cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  UPGraphcupgr 26020 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-1st 7210  df-2nd 7211  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-subgr 26205 This theorem is referenced by: (None)
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