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Theorem vtxval 28971

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), (1^st ‘𝐺), (Base‘𝐺))

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

Step	Hyp	Ref	Expression
1		eleq1 2817	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6817	. . . 4 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
3		fveq2 6817	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4	1, 2, 3	ifbieq12d 4502	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
5		df-vtx 28969	. . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
6		fvex 6830	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
7		fvex 6830	. . . 4 ⊢ (Base‘𝐺) ∈ V
8	6, 7	ifex 4524	. . 3 ⊢ if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 6924	. 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
10		fvprc 6809	. . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
11		prcnel 3460	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4484	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺))
13		fvprc 6809	. . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2776	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
15	9, 14	pm2.61i 182	1 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2110 Vcvv 3434 ∅c0 4281 ifcif 4473 × cxp 5612 ‘cfv 6477 1^st c1st 7914 Basecbs 17112 Vtxcvtx 28967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3394 df-v 3436 df-dif 3903 df-un 3905 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-iota 6433 df-fun 6479 df-fv 6485 df-vtx 28969

This theorem is referenced by: opvtxval 28974 funvtxdmge2val 28982 funvtxdm2val 28984 snstrvtxval 29008 vtxval0 29010