vtxvalg - Intuitionistic Logic Explorer

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Theorem vtxvalg 16171

Description: The set of vertices of a graph. (Contributed by AV, 9-Jan-2020.) (Revised by AV, 21-Sep-2020.)

**Assertion**
Ref	Expression
vtxvalg	⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))

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

Step	Hyp	Ref	Expression
1		df-vtx 16169	. 2 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
2		eleq1 2301	. . 3 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 5690	. . 3 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
4		fveq2 5690	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 3664	. 2 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
6		elex 2833	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		1stexg 6391	. . 3 ⊢ (𝐺 ∈ 𝑉 → (1^st ‘𝐺) ∈ V)
8		basfn 13389	. . . 4 ⊢ Base Fn V
9		funfvex 5707	. . . . 5 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
10	9	funfni 5478	. . . 4 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
11	8, 6, 10	sylancr 418	. . 3 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
12	7, 11	ifexd 4625	. 2 ⊢ (𝐺 ∈ 𝑉 → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V)
13	1, 5, 6, 12	fvmptd3 5793	1 ⊢ (𝐺 ∈ 𝑉 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1402 ∈ wcel 2209 Vcvv 2821 ifcif 3635 × cxp 4767 Fn wfn 5367 ‘cfv 5372 1^st c1st 6362 Basecbs 13330 Vtxcvtx 16167

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-cnex 8260 ax-resscn 8261 ax-1re 8263 ax-addrcl 8266

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-un 3224 df-in 3226 df-ss 3233 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fo 5378 df-fv 5380 df-1st 6364 df-inn 9284 df-ndx 13333 df-slot 13334 df-base 13336 df-vtx 16169

This theorem is referenced by: vtxex 16173 opvtxval 16176 funvtxdm2domval 16184 funvtxdm2vald 16186 vtxval0 16208