vtxvalg - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > vtxvalg		GIF version

Theorem vtxvalg 15690

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 15688	. 2 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
2		eleq1 2269	. . 3 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 5589	. . 3 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
4		fveq2 5589	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 3602	. 2 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
6		elex 2785	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		1stexg 6266	. . 3 ⊢ (𝐺 ∈ 𝑉 → (1^st ‘𝐺) ∈ V)
8		basfn 12965	. . . 4 ⊢ Base Fn V
9		funfvex 5606	. . . . 5 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
10	9	funfni 5385	. . . 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 4539	. 2 ⊢ (𝐺 ∈ 𝑉 → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V)
13	1, 5, 6, 12	fvmptd3 5686	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 1373 ∈ wcel 2177 Vcvv 2773 ifcif 3575 × cxp 4681 Fn wfn 5275 ‘cfv 5280 1^st c1st 6237 Basecbs 12907 Vtxcvtx 15686

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-cnex 8036 ax-resscn 8037 ax-1re 8039 ax-addrcl 8042

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-un 3174 df-in 3176 df-ss 3183 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-br 4052 df-opab 4114 df-mpt 4115 df-id 4348 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-fo 5286 df-fv 5288 df-1st 6239 df-inn 9057 df-ndx 12910 df-slot 12911 df-base 12913 df-vtx 15688

This theorem is referenced by: vtxex 15692 opvtxval 15695 funvtxdm2domval 15703 funvtxdm2vald 15705 vtxval0 15725