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Theorem vtxval 29147

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 2849	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6863	. . . 4 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
3		fveq2 6863	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4	1, 2, 3	ifbieq12d 4508	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
5		df-vtx 29145	. . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
6		fvex 6876	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
7		fvex 6876	. . . 4 ⊢ (Base‘𝐺) ∈ V
8	6, 7	ifex 4530	. . 3 ⊢ if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 6971	. 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
10		fvprc 6855	. . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
11		prcnel 3478	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4490	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺))
13		fvprc 6855	. . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2807	. 2 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
15	9, 14	pm2.61i 183	1 ⊢ (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ∅c0 4285 ifcif 4479 × cxp 5643 ‘cfv 6517 1^st c1st 7964 Basecbs 17228 Vtxcvtx 29143

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-iota 6473 df-fun 6519 df-fv 6525 df-vtx 29145

This theorem is referenced by: opvtxval 29150 funvtxdmge2val 29158 funvtxdm2val 29160 snstrvtxval 29184 vtxval0 29186