opvtxfv - Metamath Proof Explorer

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Theorem opvtxfv 29447

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 5700	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 29446	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 18	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = (1^st ‘⟨𝑉, 𝐸⟩))
4		op1stg 8001	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2797	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ⟨cop 4593 × cxp 5657 ‘cfv 6537 1^st c1st 7987 Vtxcvtx 29439

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fv 6545 df-1st 7989 df-vtx 29441

This theorem is used by: opvtxov 29448 opvtxfvi 29452 gropd 29474 isuhgrop 29513 uhgrunop 29518 upgrop 29537 upgr1eop 29558 upgrunop 29562 umgrunop 29564 isuspgrop 29607 isusgrop 29608 ausgrusgrb 29611 uspgr1eop 29693 usgr1eop 29696 usgrexmpllem 29706 uhgrspan1lem2 29747 upgrres1lem2 29757 opfusgr 29769 fusgrfisbase 29774 fusgrfisstep 29775 usgrexi 29887 cusgrexi 29889 p1evtxdeqlem 29958 p1evtxdeq 29959 p1evtxdp1 29960 uspgrloopvtx 29961 umgr2v2evtx 29967 wlk2v2e 30623 eupthvdres 30701 eupth2lemb 30703 konigsbergvtx 30712 konigsberg 30723 isubgrvtx 48770 opstrgric 48829 ushggricedg 48830 usgrexmpl1vtx 48926 usgrexmpl2vtx 48931 uspgrsprfo 49051

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