opvtxfv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opvtxfv		Structured version Visualization version GIF version

Theorem opvtxfv 28984

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 21-Sep-2020.)

**Assertion**
Ref	Expression
opvtxfv	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

**Proof of Theorem opvtxfv**
Step	Hyp	Ref	Expression
1		opelvvg 5672	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 28983	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 17	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
4		op1stg 7959	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2764	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3444 ⟨cop 4591 × cxp 5629 ‘cfv 6499 1^st c1st 7945 Vtxcvtx 28976

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 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

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 3403 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6452 df-fun 6501 df-fv 6507 df-1st 7947 df-vtx 28978

This theorem is referenced by: opvtxov 28985 opvtxfvi 28989 gropd 29011 isuhgrop 29050 uhgrunop 29055 upgrop 29074 upgr1eop 29095 upgrunop 29099 umgrunop 29101 isuspgrop 29141 isusgrop 29142 ausgrusgrb 29145 uspgr1eop 29227 usgr1eop 29230 usgrexmpllem 29240 uhgrspan1lem2 29281 upgrres1lem2 29291 opfusgr 29303 fusgrfisbase 29308 fusgrfisstep 29309 usgrexi 29421 cusgrexi 29423 p1evtxdeqlem 29493 p1evtxdeq 29494 p1evtxdp1 29495 uspgrloopvtx 29496 umgr2v2evtx 29502 wlk2v2e 30136 eupthvdres 30214 eupth2lemb 30216 konigsbergvtx 30225 konigsberg 30236 isubgrvtx 47860 opstrgric 47919 ushggricedg 47920 usgrexmpl1vtx 48007 usgrexmpl2vtx 48012 uspgrsprfo 48129

W3C validator