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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 36962 | . 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴) | |
2 | rncnvepres 36810 | . 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴 | |
3 | 1, 2 | eqtri 2761 | 1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cuni 4866 E cep 5537 ◡ccnv 5633 dom cdm 5634 ran crn 5635 ↾ cres 5636 ≀ ccoss 36680 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-eprel 5538 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-coss 36919 |
This theorem is referenced by: dmcoels 36965 |
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