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 29207

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 5690	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 29206	. . 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 7984	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2799	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1562 ∈ wcel 2144 Vcvv 3456 ⟨cop 4590 × cxp 5647 ‘cfv 6523 1^st c1st 7970 Vtxcvtx 29199

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-sep 5248 ax-nul 5258 ax-pr 5392 ax-un 7720

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3079 df-rex 3089 df-rab 3417 df-v 3458 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5544 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-iota 6479 df-fun 6525 df-fv 6531 df-1st 7972 df-vtx 29201

This theorem is referenced by: opvtxov 29208 opvtxfvi 29212 gropd 29234 isuhgrop 29273 uhgrunop 29278 upgrop 29297 upgr1eop 29318 upgrunop 29322 umgrunop 29324 isuspgrop 29364 isusgrop 29365 ausgrusgrb 29368 uspgr1eop 29450 usgr1eop 29453 usgrexmpllem 29463 uhgrspan1lem2 29504 upgrres1lem2 29514 opfusgr 29526 fusgrfisbase 29531 fusgrfisstep 29532 usgrexi 29644 cusgrexi 29646 p1evtxdeqlem 29715 p1evtxdeq 29716 p1evtxdp1 29717 uspgrloopvtx 29718 umgr2v2evtx 29724 wlk2v2e 30361 eupthvdres 30439 eupth2lemb 30441 konigsbergvtx 30450 konigsberg 30461 isubgrvtx 48494 opstrgric 48553 ushggricedg 48554 usgrexmpl1vtx 48650 usgrexmpl2vtx 48655 uspgrsprfo 48775

W3C validator