Theorem dmcoels 38717

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 38672	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
2	1	dmeqi 5852	. 2 ⊢ dom ∼ 𝐴 = dom ≀ (◡ E ↾ 𝐴)
3		dm1cosscnvepres 38716	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴
4	2, 3	eqtri 2758	1 ⊢ dom ∼ 𝐴 = ∪ 𝐴

Syntax hints:   = wceq 1542  ∪ cuni 4862   E cep 5522  ^◡ccnv 5622  dom cdm 5623   ↾ cres 5625   ≀ ccoss 38353   ∼ ccoels 38354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-eprel 5523  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-coss 38671  df-coels 38672

This theorem is referenced by:  dmqscoelseq  38916