vtxvalOLD - Metamath Proof Explorer

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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	⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))

Proof of Theorem vtxvalOLD Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3364	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
2		eleq1 2838	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 6333	. . . 4 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
4		fveq2 6333	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 4253	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
6		df-vtx 26097	. . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
7		fvex 6344	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
8		fvex 6344	. . . 4 ⊢ (Base‘𝐺) ∈ V
9	7, 8	ifex 4296	. . 3 ⊢ if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V
10	5, 6, 9	fvmpt 6426	. 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
11	1, 10	syl 17	1 ⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ifcif 4226 × cxp 5248 ‘cfv 6030 1^st 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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