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Theorem vtxval 25677
 Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
Assertion
Ref Expression
vtxval (𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))

Proof of Theorem vtxval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐺𝑉𝐺 ∈ V)
2 eleq1 2676 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
4 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
52, 3, 4ifbieq12d 4063 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
6 df-vtx 25675 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
7 fvex 6113 . . . 4 (1st𝐺) ∈ V
8 fvex 6113 . . . 4 (Base‘𝐺) ∈ V
97, 8ifex 4106 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
105, 6, 9fvmpt 6191 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
111, 10syl 17 1 (𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   × cxp 5036  ‘cfv 5804  1st c1st 7057  Basecbs 15695  Vtxcvtx 25673 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-vtx 25675 This theorem is referenced by:  opvtxval  25680  funvtxdm2val  25688  funvtxdmge2val  25691  snstrvtxval  25712  vtxval0  25714  vtxvalsnop  25716
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