Theorem dmcoels 38448

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 38403	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
2	1	dmeqi 5868	. 2 ⊢ dom ∼ 𝐴 = dom ≀ (◡ E ↾ 𝐴)
3		dm1cosscnvepres 38447	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴
4	2, 3	eqtri 2752	1 ⊢ dom ∼ 𝐴 = ∪ 𝐴

Syntax hints:   = wceq 1540  ∪ cuni 4871   E cep 5537  ^◡ccnv 5637  dom cdm 5638   ↾ cres 5640   ≀ ccoss 38169   ∼ ccoels 38170

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-coss 38402  df-coels 38403

This theorem is referenced by:  dmqscoelseq  38653