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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmcoels | Structured version Visualization version GIF version |
Description: The domain of coelements in 𝐴 is the union of 𝐴. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Peter Mazsa, 5-Apr-2018.) (Revised by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
dmcoels | ⊢ dom ∼ 𝐴 = ∪ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-coels 35675 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
2 | 1 | dmeqi 5773 | . 2 ⊢ dom ∼ 𝐴 = dom ≀ (◡ E ↾ 𝐴) |
3 | dm1cosscnvepres 35711 | . 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴 | |
4 | 2, 3 | eqtri 2844 | 1 ⊢ dom ∼ 𝐴 = ∪ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cuni 4838 E cep 5464 ◡ccnv 5554 dom cdm 5555 ↾ cres 5557 ≀ ccoss 35468 ∼ ccoels 35469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-eprel 5465 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-coss 35674 df-coels 35675 |
This theorem is referenced by: dmqscoelseq 35910 |
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