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Theorem opvtxval 25864
 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 25859 . 2 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
2 iftrue 4083 . 2 (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (1st𝐺))
31, 2syl5eq 2666 1 (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1481   ∈ wcel 1988  Vcvv 3195  ifcif 4077   × cxp 5102  ‘cfv 5876  1st c1st 7151  Basecbs 15838  Vtxcvtx 25855 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-vtx 25857 This theorem is referenced by:  opvtxfv  25865
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