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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 5717 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → ⟨𝑉, 𝐸⟩ ∈ (V × V)) | |
2 | opiedgval 28304 | . . 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 7990 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘⟨𝑉, 𝐸⟩) = 𝐸) | |
5 | 3, 4 | eqtrd 2772 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4634 × cxp 5674 ‘cfv 6543 2nd c2nd 7976 iEdgciedg 28295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-2nd 7978 df-iedg 28297 |
This theorem is referenced by: opiedgov 28306 opiedgfvi 28308 gropd 28329 edgopval 28349 isuhgrop 28368 uhgrunop 28373 upgrop 28392 upgr0eop 28412 upgr1eop 28413 upgrunop 28417 umgrunop 28419 isuspgrop 28459 isusgrop 28460 ausgrusgrb 28463 usgr0eop 28541 uspgr1eop 28542 usgr1eop 28545 usgrexmpllem 28555 uhgrspan1lem3 28597 upgrres1lem3 28607 fusgrfisbase 28623 fusgrfisstep 28624 usgrexi 28736 cusgrexi 28738 p1evtxdeqlem 28807 p1evtxdeq 28808 p1evtxdp1 28809 uspgrloopiedg 28812 umgr2v2eiedg 28818 wlk2v2e 29448 eupthvdres 29526 eupth2lemb 29528 konigsbergiedg 29538 strisomgrop 46587 ushrisomgr 46588 |
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