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Theorem vtxval 26099
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 2838 . . . 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 4252 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
5 df-vtx 26097 . . 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 4295 . . 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 3369 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4236 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (Base‘𝐺))
13 fvprc 6326 . . 3 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2816 . 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 1631  wcel 2145  Vcvv 3351  c0 4063  ifcif 4225   × cxp 5247  cfv 6031  1st c1st 7313  Basecbs 16064  Vtxcvtx 26095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  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 26097
This theorem is referenced by:  opvtxval  26104  funvtxdmge2val  26112  funvtxdm2val  26114  snstrvtxval  26150  vtxval0  26152
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