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Theorem vtxval 28963

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 2816	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
2		fveq2 6826	. . . 4 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
3		fveq2 6826	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4	1, 2, 3	ifbieq12d 4507	. . 3 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
5		df-vtx 28961	. . 3 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
6		fvex 6839	. . . 4 ⊢ (1^st ‘𝐺) ∈ V
7		fvex 6839	. . . 4 ⊢ (Base‘𝐺) ∈ V
8	6, 7	ifex 4529	. . 3 ⊢ if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V
9	4, 5, 8	fvmpt 6934	. 2 ⊢ (𝐺 ∈ V → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
10		fvprc 6818	. . 3 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
11		prcnel 3464	. . . 4 ⊢ (¬ 𝐺 ∈ V → ¬ 𝐺 ∈ (V × V))
12	11	iffalsed 4489	. . 3 ⊢ (¬ 𝐺 ∈ V → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺))
13		fvprc 6818	. . 3 ⊢ (¬ 𝐺 ∈ V → (Vtx‘𝐺) = ∅)
14	10, 12, 13	3eqtr4rd 2775	. 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 1540 ∈ wcel 2109 Vcvv 3438 ∅c0 4286 ifcif 4478 × cxp 5621 ‘cfv 6486 1^st c1st 7929 Basecbs 17138 Vtxcvtx 28959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3397 df-v 3440 df-dif 3908 df-un 3910 df-ss 3922 df-nul 4287 df-if 4479 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-iota 6442 df-fun 6488 df-fv 6494 df-vtx 28961

This theorem is referenced by: opvtxval 28966 funvtxdmge2val 28974 funvtxdm2val 28976 snstrvtxval 29000 vtxval0 29002