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| Mirrors > Home > MPE Home > Th. List > opiedgfv | Structured version Visualization version GIF version | ||
| 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.) |
| Ref | Expression |
|---|---|
| opiedgfv | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | opelvvg 5726 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 〈𝑉, 𝐸〉 ∈ (V × V)) | |
| 2 | opiedgval 29023 | . . 3 ⊢ (〈𝑉, 𝐸〉 ∈ (V × V) → (iEdg‘〈𝑉, 𝐸〉) = (2nd ‘〈𝑉, 𝐸〉)) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = (2nd ‘〈𝑉, 𝐸〉)) |
| 4 | op2ndg 8027 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘〈𝑉, 𝐸〉) = 𝐸) | |
| 5 | 3, 4 | eqtrd 2777 | 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 3480 〈cop 4632 × cxp 5683 ‘cfv 6561 2nd c2nd 8013 iEdgciedg 29014 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-2nd 8015 df-iedg 29016 |
| This theorem is referenced by: opiedgov 29025 opiedgfvi 29027 gropd 29048 edgopval 29068 isuhgrop 29087 uhgrunop 29092 upgrop 29111 upgr0eop 29131 upgr1eop 29132 upgrunop 29136 umgrunop 29138 isuspgrop 29178 isusgrop 29179 ausgrusgrb 29182 usgr0eop 29263 uspgr1eop 29264 usgr1eop 29267 usgrexmpllem 29277 uhgrspan1lem3 29319 upgrres1lem3 29329 fusgrfisbase 29345 fusgrfisstep 29346 usgrexi 29458 cusgrexi 29460 p1evtxdeqlem 29530 p1evtxdeq 29531 p1evtxdp1 29532 uspgrloopiedg 29535 umgr2v2eiedg 29541 wlk2v2e 30176 eupthvdres 30254 eupth2lemb 30256 konigsbergiedg 30266 isubgriedg 47849 opstrgric 47895 ushggricedg 47896 usgrexmpl1edg 47983 usgrexmpl2edg 47988 |
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