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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 28817 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | edgiedgb.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | eqcomi 2735 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
4 | 3 | rneqi 5930 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
5 | 1, 4 | eqtri 2754 | . . 3 ⊢ (Edg‘𝐺) = ran 𝐼 |
6 | 5 | eleq2i 2819 | . 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼) |
7 | elrnrexdmb 7085 | . 2 ⊢ (Fun 𝐼 → (𝐸 ∈ ran 𝐼 ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼‘𝑥))) | |
8 | 6, 7 | bitrid 283 | 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 3064 dom cdm 5669 ran crn 5670 Fun wfun 6531 ‘cfv 6537 iEdgciedg 28765 Edgcedg 28815 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-fv 6545 df-edg 28816 |
This theorem is referenced by: uhgredgiedgb 28894 |
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