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Theorem dmcoels 36965
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
StepHypRef Expression
1 df-coels 36920 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
21dmeqi 5861 . 2 dom ∼ 𝐴 = dom ≀ ( E ↾ 𝐴)
3 dm1cosscnvepres 36964 . 2 dom ≀ ( E ↾ 𝐴) = 𝐴
42, 3eqtri 2761 1 dom ∼ 𝐴 = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   cuni 4866   E cep 5537  ccnv 5633  dom cdm 5634  cres 5636  ccoss 36680  ccoels 36681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-eprel 5538  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-coss 36919  df-coels 36920
This theorem is referenced by:  dmqscoelseq  37169
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