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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 27886 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | edgiedgb.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | eqcomi 2745 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
4 | 3 | rneqi 5890 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
5 | 1, 4 | eqtri 2764 | . . 3 ⊢ (Edg‘𝐺) = ran 𝐼 |
6 | 5 | eleq2i 2829 | . 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼) |
7 | elrnrexdmb 7036 | . 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 1541 ∈ wcel 2106 ∃wrex 3071 dom cdm 5631 ran crn 5632 Fun wfun 6487 ‘cfv 6493 iEdgciedg 27834 Edgcedg 27884 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6445 df-fun 6495 df-fn 6496 df-fv 6501 df-edg 27885 |
This theorem is referenced by: uhgredgiedgb 27963 |
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