Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6846 | . 2 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ ran 𝐸) | |
2 | edgval 26836 | . . 3 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | iedgedg.e | . . . 4 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | 3 | rneqi 5809 | . . 3 ⊢ ran 𝐸 = ran (iEdg‘𝐺) |
5 | 2, 4 | eqtr4i 2849 | . 2 ⊢ (Edg‘𝐺) = ran 𝐸 |
6 | 1, 5 | eleqtrrdi 2926 | 1 ⊢ ((Fun 𝐸 ∧ 𝐼 ∈ dom 𝐸) → (𝐸‘𝐼) ∈ (Edg‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 dom cdm 5557 ran crn 5558 Fun wfun 6351 ‘cfv 6357 iEdgciedg 26784 Edgcedg 26834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-edg 26835 |
This theorem is referenced by: edglnl 26930 numedglnl 26931 umgr2cycllem 32389 |
Copyright terms: Public domain | W3C validator |