opiedgfv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opiedgfv		Structured version Visualization version GIF version

Theorem opiedgfv 29143

Description: The set of indexed edges 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
opiedgfv	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

**Proof of Theorem opiedgfv**
Step	Hyp	Ref	Expression
1		opelvvg 5677	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 29142	. . 3 ⊢ (⟨𝑉, 𝐸⟩ ∈ (V × V) → (iEdg‘⟨𝑉, 𝐸⟩) = (2^nd ‘⟨𝑉, 𝐸⟩))
3	1, 2	syl 17	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = (2^nd ‘⟨𝑉, 𝐸⟩))
4		op2ndg 7968	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2^nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2787	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1550 ∈ wcel 2132 Vcvv 3444 ⟨cop 4578 × cxp 5634 ‘cfv 6506 2^nd c2nd 7954 iEdgciedg 29133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-sep 5236 ax-nul 5246 ax-pr 5380 ax-un 7703

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-rab 3405 df-v 3446 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-iota 6462 df-fun 6508 df-fv 6514 df-2nd 7956 df-iedg 29135

This theorem is referenced by: opiedgov 29144 opiedgfvi 29146 gropd 29167 edgopval 29187 isuhgrop 29206 uhgrunop 29211 upgrop 29230 upgr0eop 29250 upgr1eop 29251 upgrunop 29255 umgrunop 29257 isuspgrop 29297 isusgrop 29298 ausgrusgrb 29301 usgr0eop 29382 uspgr1eop 29383 usgr1eop 29386 usgrexmpllem 29396 uhgrspan1lem3 29438 upgrres1lem3 29448 fusgrfisbase 29464 fusgrfisstep 29465 usgrexi 29577 cusgrexi 29579 p1evtxdeqlem 29648 p1evtxdeq 29649 p1evtxdp1 29650 uspgrloopiedg 29653 umgr2v2eiedg 29659 wlk2v2e 30294 eupthvdres 30372 eupth2lemb 30374 konigsbergiedg 30384 isubgriedg 48423 opstrgric 48486 ushggricedg 48487 usgrexmpl1edg 48584 usgrexmpl2edg 48589

W3C validator