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Theorem opvtxval 29035
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 29032 . 2 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
2 iftrue 4537 . 2 (𝐺 ∈ (V × V) → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (1st𝐺))
31, 2eqtrid 2787 1 (𝐺 ∈ (V × V) → (Vtx‘𝐺) = (1st𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  ifcif 4531   × cxp 5687  cfv 6563  1st c1st 8011  Basecbs 17245  Vtxcvtx 29028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-vtx 29030
This theorem is referenced by:  opvtxfv  29036
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