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Theorem vtxval 28525
Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)
Assertion
Ref Expression
vtxval (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 eleq1 2819 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6892 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
3 fveq2 6892 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
41, 2, 3ifbieq12d 4557 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
5 df-vtx 28523 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
6 fvex 6905 . . . 4 (1st𝐺) ∈ V
7 fvex 6905 . . . 4 (Base‘𝐺) ∈ V
86, 7ifex 4579 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
94, 5, 8fvmpt 6999 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
10 fvprc 6884 . . 3 𝐺 ∈ V → (Base‘𝐺) = ∅)
11 prcnel 3496 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4540 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (Base‘𝐺))
13 fvprc 6884 . . 3 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2781 . 2 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
159, 14pm2.61i 182 1 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2104  Vcvv 3472  c0 4323  ifcif 4529   × cxp 5675  cfv 6544  1st c1st 7977  Basecbs 17150  Vtxcvtx 28521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-vtx 28523
This theorem is referenced by:  opvtxval  28528  funvtxdmge2val  28536  funvtxdm2val  28538  snstrvtxval  28562  vtxval0  28564
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