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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dm1cosscnvepres | Structured version Visualization version GIF version | ||
| Description: The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.) |
| Ref | Expression |
|---|---|
| dm1cosscnvepres | ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmcoss2 38455 | . 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴) | |
| 2 | rncnvepres 38304 | . 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 | |
| 3 | 1, 2 | eqtri 2765 | 1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cuni 4907 E cep 5583 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 ≀ ccoss 38182 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-coss 38412 |
| This theorem is referenced by: dmcoels 38458 |
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