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Theorem opvtxfv 25681

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 5087	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 25680	. . 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 7071	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2644	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟨cop 4131 × cxp 5036 ‘cfv 5804 1^st c1st 7057 Vtxcvtx 25673

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059 df-vtx 25675

This theorem is referenced by: opvtxov 25682 opvtxfvi 25686 graop 25706 gropd 25708 isuhgrop 25736 uhgrunop 25741 upgr1eop 25781 upgrunop 25785 umgrunop 25787 isusgrop 40392 ausgrusgrb 40395 uspgr1eop 40473 usgr1eop 40476 usgrexmpllem 40484 uhgrspanop 40520 uhgrspan1lem2 40525 upgrres1lem2 40530 opfusgr 40542 fusgrfisbase 40547 fusgrfisstep 40548 usgrexi 40661 cusgrexi 40662 cusgrsize 40670 fusgrmaxsize 40680 p1evtxdeqlem 40728 p1evtxdeq 40729 p1evtxdp1 40730 uspgrloopvtx 40731 umgr2v2evtx 40737 1wlk2v2e 41324 eupthvdres 41403 eupth2lemb 41405 konigsbergvtx 41414 konigsberg-av 41427

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