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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 5730 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 〈𝑉, 𝐸〉 ∈ (V × V)) | |
2 | opiedgval 29038 | . . 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 8026 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘〈𝑉, 𝐸〉) = 𝐸) | |
5 | 3, 4 | eqtrd 2775 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 〈cop 4637 × cxp 5687 ‘cfv 6563 2nd c2nd 8012 iEdgciedg 29029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-2nd 8014 df-iedg 29031 |
This theorem is referenced by: opiedgov 29040 opiedgfvi 29042 gropd 29063 edgopval 29083 isuhgrop 29102 uhgrunop 29107 upgrop 29126 upgr0eop 29146 upgr1eop 29147 upgrunop 29151 umgrunop 29153 isuspgrop 29193 isusgrop 29194 ausgrusgrb 29197 usgr0eop 29278 uspgr1eop 29279 usgr1eop 29282 usgrexmpllem 29292 uhgrspan1lem3 29334 upgrres1lem3 29344 fusgrfisbase 29360 fusgrfisstep 29361 usgrexi 29473 cusgrexi 29475 p1evtxdeqlem 29545 p1evtxdeq 29546 p1evtxdp1 29547 uspgrloopiedg 29550 umgr2v2eiedg 29556 wlk2v2e 30186 eupthvdres 30264 eupth2lemb 30266 konigsbergiedg 30276 isubgriedg 47787 opstrgric 47833 ushggricedg 47834 usgrexmpl1edg 47919 usgrexmpl2edg 47924 |
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