Theorem dmcoels 38927

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.)

Assertion

Ref	Expression
dmcoels	⊢ dom ∼ 𝐴 = ∪ 𝐴

Proof of Theorem dmcoels

Step	Hyp	Ref	Expression
1		df-coels 38882	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
2	1	dmeqi 5852	. 2 ⊢ dom ∼ 𝐴 = dom ≀ (◡ E ↾ 𝐴)
3		dm1cosscnvepres 38926	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴
4	2, 3	eqtri 2764	1 ⊢ dom ∼ 𝐴 = ∪ 𝐴

Syntax hints:   = wceq 1548  ∪ cuni 4840   E cep 5519  ^◡ccnv 5619  dom cdm 5620   ↾ cres 5622   ≀ ccoss 38563   ∼ ccoels 38564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-eprel 5520  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-coss 38881  df-coels 38882

This theorem is referenced by:  dmqscoelseq  39126