Theorem dmcoels 39146

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 39101	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
2	1	dmeqi 5898	. 2 ⊢ dom ∼ 𝐴 = dom ≀ (◡ E ↾ 𝐴)
3		dm1cosscnvepres 39145	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴
4	2, 3	eqtri 2793	1 ⊢ dom ∼ 𝐴 = ∪ 𝐴

Syntax hints:   = wceq 1568  ∪ cuni 4877   E cep 5564  ^◡ccnv 5664  dom cdm 5665   ↾ cres 5667   ≀ ccoss 38782   ∼ ccoels 38783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-eprel 5565  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-coss 39100  df-coels 39101

This theorem is referenced by:  dmqscoelseq  39345