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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 26828 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | edgiedgb.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | eqcomi 2830 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
4 | 3 | rneqi 5801 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
5 | 1, 4 | eqtri 2844 | . . 3 ⊢ (Edg‘𝐺) = ran 𝐼 |
6 | 5 | eleq2i 2904 | . 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼) |
7 | elrnrexdmb 6850 | . 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) | |
8 | 6, 7 | syl5bb 285 | 1 ⊢ (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 dom cdm 5549 ran crn 5550 Fun wfun 6343 ‘cfv 6349 iEdgciedg 26776 Edgcedg 26826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-fv 6357 df-edg 26827 |
This theorem is referenced by: uhgredgiedgb 26905 |
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