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Theorem vtxval 28125
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 2820 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6878 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
3 fveq2 6878 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
41, 2, 3ifbieq12d 4550 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
5 df-vtx 28123 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
6 fvex 6891 . . . 4 (1st𝐺) ∈ V
7 fvex 6891 . . . 4 (Base‘𝐺) ∈ V
86, 7ifex 4572 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
94, 5, 8fvmpt 6984 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
10 fvprc 6870 . . 3 𝐺 ∈ V → (Base‘𝐺) = ∅)
11 prcnel 3496 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4533 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (Base‘𝐺))
13 fvprc 6870 . . 3 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2782 . 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 1541  wcel 2106  Vcvv 3473  c0 4318  ifcif 4522   × cxp 5667  cfv 6532  1st c1st 7955  Basecbs 17126  Vtxcvtx 28121
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-vtx 28123
This theorem is referenced by:  opvtxval  28128  funvtxdmge2val  28136  funvtxdm2val  28138  snstrvtxval  28162  vtxval0  28164
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