vtxvalg - Intuitionistic Logic Explorer

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Theorem vtxvalg 15557

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 15555	. 2 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
2		eleq1 2267	. . 3 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 5575	. . 3 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
4		fveq2 5575	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 3596	. 2 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
6		elex 2782	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		1stexg 6252	. . 3 ⊢ (𝐺 ∈ 𝑉 → (1^st ‘𝐺) ∈ V)
8		basfn 12832	. . . 4 ⊢ Base Fn V
9		funfvex 5592	. . . . 5 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
10	9	funfni 5375	. . . 4 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
11	8, 6, 10	sylancr 414	. . 3 ⊢ (𝐺 ∈ 𝑉 → (Base‘𝐺) ∈ V)
12	7, 11	ifexd 4530	. 2 ⊢ (𝐺 ∈ 𝑉 → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V)
13	1, 5, 6, 12	fvmptd3 5672	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 1372 ∈ wcel 2175 Vcvv 2771 ifcif 3570 × cxp 4672 Fn wfn 5265 ‘cfv 5270 1^st c1st 6223 Basecbs 12774 Vtxcvtx 15553

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-cnex 8015 ax-resscn 8016 ax-1re 8018 ax-addrcl 8021

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-un 3169 df-in 3171 df-ss 3178 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-int 3885 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-fo 5276 df-fv 5278 df-1st 6225 df-inn 9036 df-ndx 12777 df-slot 12778 df-base 12780 df-vtx 15555

This theorem is referenced by: opvtxval 15560 funvtxdm2domval 15568 funvtxdm2vald 15570