opvtxfv - Metamath Proof Explorer

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Theorem opvtxfv 28295

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 5718	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 28294	. . 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 7987	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2773	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⟨cop 4635 × cxp 5675 ‘cfv 6544 1^st c1st 7973 Vtxcvtx 28287

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-vtx 28289

This theorem is referenced by: opvtxov 28296 opvtxfvi 28300 gropd 28322 isuhgrop 28361 uhgrunop 28366 upgrop 28385 upgr1eop 28406 upgrunop 28410 umgrunop 28412 isuspgrop 28452 isusgrop 28453 ausgrusgrb 28456 uspgr1eop 28535 usgr1eop 28538 usgrexmpllem 28548 uhgrspan1lem2 28589 upgrres1lem2 28599 opfusgr 28611 fusgrfisbase 28616 fusgrfisstep 28617 usgrexi 28729 cusgrexi 28731 p1evtxdeqlem 28800 p1evtxdeq 28801 p1evtxdp1 28802 uspgrloopvtx 28803 umgr2v2evtx 28809 wlk2v2e 29441 eupthvdres 29519 eupth2lemb 29521 konigsbergvtx 29530 konigsberg 29541 strisomgrop 46556 ushrisomgr 46557 uspgrsprfo 46574

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