![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5559 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 〈𝑉, 𝐸〉 ∈ (V × V)) | |
2 | opiedgval 26799 | . . 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 7684 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (2nd ‘〈𝑉, 𝐸〉) = 𝐸) | |
5 | 3, 4 | eqtrd 2833 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘〈𝑉, 𝐸〉) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 ‘cfv 6324 2nd c2nd 7670 iEdgciedg 26790 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-2nd 7672 df-iedg 26792 |
This theorem is referenced by: opiedgov 26801 opiedgfvi 26803 gropd 26824 edgopval 26844 isuhgrop 26863 uhgrunop 26868 upgrop 26887 upgr0eop 26907 upgr1eop 26908 upgrunop 26912 umgrunop 26914 isuspgrop 26954 isusgrop 26955 ausgrusgrb 26958 usgr0eop 27036 uspgr1eop 27037 usgr1eop 27040 usgrexmpllem 27050 uhgrspan1lem3 27092 upgrres1lem3 27102 fusgrfisbase 27118 fusgrfisstep 27119 usgrexi 27231 cusgrexi 27233 p1evtxdeqlem 27302 p1evtxdeq 27303 p1evtxdp1 27304 uspgrloopiedg 27307 umgr2v2eiedg 27313 wlk2v2e 27942 eupthvdres 28020 eupth2lemb 28022 konigsbergiedg 28032 strisomgrop 44358 ushrisomgr 44359 |
Copyright terms: Public domain | W3C validator |