vtxvalg - Intuitionistic Logic Explorer

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

Theorem vtxvalg 15811

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 15809	. 2 ⊢ Vtx = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)))
2		eleq1 2292	. . 3 ⊢ (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3		fveq2 5626	. . 3 ⊢ (𝑔 = 𝐺 → (1^st ‘𝑔) = (1^st ‘𝐺))
4		fveq2 5626	. . 3 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5	2, 3, 4	ifbieq12d 3629	. 2 ⊢ (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (1^st ‘𝑔), (Base‘𝑔)) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
6		elex 2811	. 2 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
7		1stexg 6311	. . 3 ⊢ (𝐺 ∈ 𝑉 → (1^st ‘𝐺) ∈ V)
8		basfn 13086	. . . 4 ⊢ Base Fn V
9		funfvex 5643	. . . . 5 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
10	9	funfni 5422	. . . 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 4574	. 2 ⊢ (𝐺 ∈ 𝑉 → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) ∈ V)
13	1, 5, 6, 12	fvmptd3 5727	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 1395 ∈ wcel 2200 Vcvv 2799 ifcif 3602 × cxp 4716 Fn wfn 5312 ‘cfv 5317 1^st c1st 6282 Basecbs 13027 Vtxcvtx 15807

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-cnex 8086 ax-resscn 8087 ax-1re 8089 ax-addrcl 8092

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-un 3201 df-in 3203 df-ss 3210 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-fo 5323 df-fv 5325 df-1st 6284 df-inn 9107 df-ndx 13030 df-slot 13031 df-base 13033 df-vtx 15809

This theorem is referenced by: vtxex 15813 opvtxval 15816 funvtxdm2domval 15824 funvtxdm2vald 15826 vtxval0 15848