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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 28042 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
2 | edgiedgb.i | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | eqcomi 2746 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
4 | 3 | rneqi 5897 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
5 | 1, 4 | eqtri 2765 | . . 3 ⊢ (Edg‘𝐺) = ran 𝐼 |
6 | 5 | eleq2i 2830 | . 2 ⊢ (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran 𝐼) |
7 | elrnrexdmb 7045 | . 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 1542 ∈ wcel 2107 ∃wrex 3074 dom cdm 5638 ran crn 5639 Fun wfun 6495 ‘cfv 6501 iEdgciedg 27990 Edgcedg 28040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-fv 6509 df-edg 28041 |
This theorem is referenced by: uhgredgiedgb 28119 |
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