opvtxfv - Metamath Proof Explorer

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Theorem opvtxfv 28982

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 5655	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 28981	. . 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 7933	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2766	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 Vcvv 3436 ⟨cop 4579 × cxp 5612 ‘cfv 6481 1^st c1st 7919 Vtxcvtx 28974

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6437 df-fun 6483 df-fv 6489 df-1st 7921 df-vtx 28976

This theorem is referenced by: opvtxov 28983 opvtxfvi 28987 gropd 29009 isuhgrop 29048 uhgrunop 29053 upgrop 29072 upgr1eop 29093 upgrunop 29097 umgrunop 29099 isuspgrop 29139 isusgrop 29140 ausgrusgrb 29143 uspgr1eop 29225 usgr1eop 29228 usgrexmpllem 29238 uhgrspan1lem2 29279 upgrres1lem2 29289 opfusgr 29301 fusgrfisbase 29306 fusgrfisstep 29307 usgrexi 29419 cusgrexi 29421 p1evtxdeqlem 29491 p1evtxdeq 29492 p1evtxdp1 29493 uspgrloopvtx 29494 umgr2v2evtx 29500 wlk2v2e 30137 eupthvdres 30215 eupth2lemb 30217 konigsbergvtx 30226 konigsberg 30237 isubgrvtx 47966 opstrgric 48025 ushggricedg 48026 usgrexmpl1vtx 48122 usgrexmpl2vtx 48127 uspgrsprfo 48247

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