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Theorem opvtxval 26799
Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
Assertion
Ref Expression
opvtxval (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))

Proof of Theorem opvtxval
StepHypRef Expression
1 vtxval 26796 . 2 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
2 iftrue 4456 . 2 (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (1st𝐺))
31, 2syl5eq 2871 1 (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  ifcif 4450   × cxp 5540  cfv 6343  1st c1st 7682  Basecbs 16483  Vtxcvtx 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-vtx 26794
This theorem is referenced by:  opvtxfv  26800
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