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Theorem vtxvalOLD 26079
Description: Obsolete version of vtxval 26077 as of 11-Nov-2021. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
vtxvalOLD (𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))

Proof of Theorem vtxvalOLD
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elex 3352 . 2 (𝐺𝑉𝐺 ∈ V)
2 eleq1 2827 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3 fveq2 6352 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
4 fveq2 6352 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
52, 3, 4ifbieq12d 4257 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
6 df-vtx 26075 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
7 fvex 6362 . . . 4 (1st𝐺) ∈ V
8 fvex 6362 . . . 4 (Base‘𝐺) ∈ V
97, 8ifex 4300 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
105, 6, 9fvmpt 6444 . 2 (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
111, 10syl 17 1 (𝐺𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  ifcif 4230   × cxp 5264  cfv 6049  1st c1st 7331  Basecbs 16059  Vtxcvtx 26073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-vtx 26075
This theorem is referenced by:  funvtxdm2valOLD  26094  funvtxdmge2valOLD  26098
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