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Mirrors > Home > MPE Home > Th. List > iedgedg | Structured version Visualization version GIF version |
Description: An indexed edge is an edge. (Contributed by AV, 19-Dec-2021.) |
Ref | Expression |
---|---|
iedgedg.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
iedgedg | ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelrn 6951 | . 2 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ ran 𝐸) | |
2 | edgval 27417 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | iedgedg.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 3 | rneqi 5845 | . . 3 ⊢ ran 𝐸 = ran (iEdg‘𝐺) |
5 | 2, 4 | eqtr4i 2771 | . 2 ⊢ (Edg‘𝐺) = ran 𝐸 |
6 | 1, 5 | eleqtrrdi 2852 | 1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 dom cdm 5590 ran crn 5591 Fun wfun 6426 ‘cfv 6432 iEdgciedg 27365 Edgcedg 27415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6390 df-fun 6434 df-fn 6435 df-fv 6440 df-edg 27416 |
This theorem is referenced by: edglnl 27511 numedglnl 27512 umgr2cycllem 33098 |
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