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Theorem vtxval 26098
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 2837 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6332 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
3 fveq2 6332 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
41, 2, 3ifbieq12d 4250 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
5 df-vtx 26096 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
6 fvex 6342 . . . 4 (1st𝐺) ∈ V
7 fvex 6342 . . . 4 (Base‘𝐺) ∈ V
86, 7ifex 4293 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
94, 5, 8fvmpt 6424 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
10 fvprc 6326 . . 3 𝐺 ∈ V → (Base‘𝐺) = ∅)
11 prcnel 3367 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4234 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (Base‘𝐺))
13 fvprc 6326 . . 3 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2815 . 2 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
159, 14pm2.61i 176 1 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  ifcif 4223   × cxp 5247  cfv 6031  1st c1st 7312  Basecbs 16063  Vtxcvtx 26094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-vtx 26096
This theorem is referenced by:  opvtxval  26103  funvtxdmge2val  26111  funvtxdm2val  26113  snstrvtxval  26149  vtxval0  26151
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