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Theorem vtxval 26906
 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 2839 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2 fveq2 6663 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
3 fveq2 6663 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
41, 2, 3ifbieq12d 4451 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
5 df-vtx 26904 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
6 fvex 6676 . . . 4 (1st𝐺) ∈ V
7 fvex 6676 . . . 4 (Base‘𝐺) ∈ V
86, 7ifex 4473 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
94, 5, 8fvmpt 6764 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
10 fvprc 6655 . . 3 𝐺 ∈ V → (Base‘𝐺) = ∅)
11 prcnel 3434 . . . 4 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
1211iffalsed 4434 . . 3 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) = (Base‘𝐺))
13 fvprc 6655 . . 3 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
1410, 12, 133eqtr4rd 2804 . 2 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
159, 14pm2.61i 185 1 (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  ifcif 4423   × cxp 5526  ‘cfv 6340  1st c1st 7697  Basecbs 16555  Vtxcvtx 26902 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-vtx 26904 This theorem is referenced by:  opvtxval  26909  funvtxdmge2val  26917  funvtxdm2val  26919  snstrvtxval  26943  vtxval0  26945
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