![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > edgiedgb | Structured version Visualization version GIF version |
Description: A set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
Ref | Expression |
---|---|
edgiedgb.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
edgiedgb | ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | edgval 28890 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | edgiedgb.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | eqcomi 2737 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
4 | 3 | rneqi 5943 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
5 | 1, 4 | eqtri 2756 | . . 3 ⊢ (Edg‘𝐺) = ran 𝐼 |
6 | 5 | eleq2i 2821 | . 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼) |
7 | elrnrexdmb 7105 | . 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) | |
8 | 6, 7 | bitrid 282 | 1 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 dom cdm 5682 ran crn 5683 Fun wfun 6547 ‘cfv 6553 iEdgciedg 28838 Edgcedg 28888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 df-edg 28889 |
This theorem is referenced by: uhgredgiedgb 28967 |
Copyright terms: Public domain | W3C validator |