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Theorem vtxvalOLD 26101
Description: Obsolete version of vtxval 26099 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 3364 . 2 (𝐺𝑉𝐺 ∈ V)
2 eleq1 2838 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3 fveq2 6333 . . . 4 (𝑔 = 𝐺 → (1st𝑔) = (1st𝐺))
4 fveq2 6333 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
52, 3, 4ifbieq12d 4253 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)))
6 df-vtx 26097 . . 3 Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1st𝑔), (Base‘𝑔)))
7 fvex 6344 . . . 4 (1st𝐺) ∈ V
8 fvex 6344 . . . 4 (Base‘𝐺) ∈ V
97, 8ifex 4296 . . 3 if(𝐺 ∈ (V × V), (1st𝐺), (Base‘𝐺)) ∈ V
105, 6, 9fvmpt 6426 . 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 1631  wcel 2145  Vcvv 3351  ifcif 4226   × cxp 5248  cfv 6030  1st c1st 7317  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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-vtx 26097
This theorem is referenced by:  funvtxdm2valOLD  26116  funvtxdmge2valOLD  26120
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