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Theorem eqvreldmqs2 38203
Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 30-Dec-2024.)
Assertion
Ref Expression
eqvreldmqs2 (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))

Proof of Theorem eqvreldmqs2
StepHypRef Expression
1 df-coels 37939 . . . 4 𝐴 = ≀ ( E ↾ 𝐴)
21eqvreleqi 38130 . . 3 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
32bicomi 223 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ EqvRel ∼ 𝐴)
4 dmqs1cosscnvepreseq 38189 . 2 ((dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
53, 4anbi12i 626 1 (( EqvRel ≀ ( E ↾ 𝐴) ∧ (dom ≀ ( E ↾ 𝐴) / ≀ ( E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533   cuni 4903   E cep 5575  ccnv 5671  dom cdm 5672  cres 5674   / cqs 8720  ccoss 37704  ccoels 37705   EqvRel weqvrel 37721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8723  df-qs 8727  df-coss 37938  df-coels 37939  df-refrel 38039  df-symrel 38071  df-trrel 38101  df-eqvrel 38112
This theorem is referenced by:  cpet2  38364
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