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Theorem dmcoels 34750
 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 34713 . . 3 𝐴 = ≀ ( E ↾ 𝐴)
21dmeqi 5561 . 2 dom ∼ 𝐴 = dom ≀ ( E ↾ 𝐴)
3 dm1cosscnvepres 34749 . 2 dom ≀ ( E ↾ 𝐴) = 𝐴
42, 3eqtri 2849 1 dom ∼ 𝐴 = 𝐴
 Colors of variables: wff setvar class Syntax hints:   = wceq 1656  ∪ cuni 4660   E cep 5256  ◡ccnv 5345  dom cdm 5346   ↾ cres 5348   ≀ ccoss 34519   ∼ ccoels 34520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-eprel 5257  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-coss 34712  df-coels 34713 This theorem is referenced by: (None)
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