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 28986

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 5695	. . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V))
2		opiedgval 28985	. . 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 8001	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2^nd ‘⟨𝑉, 𝐸⟩) = 𝐸)
5	3, 4	eqtrd 2770	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ⟨cop 4607 × cxp 5652 ‘cfv 6531 2^nd c2nd 7987 iEdgciedg 28976

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-iota 6484 df-fun 6533 df-fv 6539 df-2nd 7989 df-iedg 28978

This theorem is referenced by: opiedgov 28987 opiedgfvi 28989 gropd 29010 edgopval 29030 isuhgrop 29049 uhgrunop 29054 upgrop 29073 upgr0eop 29093 upgr1eop 29094 upgrunop 29098 umgrunop 29100 isuspgrop 29140 isusgrop 29141 ausgrusgrb 29144 usgr0eop 29225 uspgr1eop 29226 usgr1eop 29229 usgrexmpllem 29239 uhgrspan1lem3 29281 upgrres1lem3 29291 fusgrfisbase 29307 fusgrfisstep 29308 usgrexi 29420 cusgrexi 29422 p1evtxdeqlem 29492 p1evtxdeq 29493 p1evtxdp1 29494 uspgrloopiedg 29497 umgr2v2eiedg 29503 wlk2v2e 30138 eupthvdres 30216 eupth2lemb 30218 konigsbergiedg 30228 isubgriedg 47876 opstrgric 47939 ushggricedg 47940 usgrexmpl1edg 48028 usgrexmpl2edg 48033

W3C validator