opvtxfv - Metamath Proof Explorer

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Theorem opvtxfv 28938

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 5682	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opvtxval 28937	. . 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 7983	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (1^st ‘⟨𝑉, 𝐸⟩) = 𝑉)
5	3, 4	eqtrd 2765	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ⟨cop 4598 × cxp 5639 ‘cfv 6514 1^st c1st 7969 Vtxcvtx 28930

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-iota 6467 df-fun 6516 df-fv 6522 df-1st 7971 df-vtx 28932

This theorem is referenced by: opvtxov 28939 opvtxfvi 28943 gropd 28965 isuhgrop 29004 uhgrunop 29009 upgrop 29028 upgr1eop 29049 upgrunop 29053 umgrunop 29055 isuspgrop 29095 isusgrop 29096 ausgrusgrb 29099 uspgr1eop 29181 usgr1eop 29184 usgrexmpllem 29194 uhgrspan1lem2 29235 upgrres1lem2 29245 opfusgr 29257 fusgrfisbase 29262 fusgrfisstep 29263 usgrexi 29375 cusgrexi 29377 p1evtxdeqlem 29447 p1evtxdeq 29448 p1evtxdp1 29449 uspgrloopvtx 29450 umgr2v2evtx 29456 wlk2v2e 30093 eupthvdres 30171 eupth2lemb 30173 konigsbergvtx 30182 konigsberg 30193 isubgrvtx 47871 opstrgric 47930 ushggricedg 47931 usgrexmpl1vtx 48018 usgrexmpl2vtx 48023 uspgrsprfo 48140

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