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Theorem eqvreldmqs2 38059
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 37795 . . . 4 𝐴 = ≀ ( E ↾ 𝐴)
21eqvreleqi 37986 . . 3 ( EqvRel ∼ 𝐴 ↔ EqvRel ≀ ( E ↾ 𝐴))
32bicomi 223 . 2 ( EqvRel ≀ ( E ↾ 𝐴) ↔ EqvRel ∼ 𝐴)
4 dmqs1cosscnvepreseq 38045 . 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 395   = wceq 1533   cuni 4902   E cep 5572  ccnv 5668  dom cdm 5669  cres 5671   / cqs 8704  ccoss 37556  ccoels 37557   EqvRel weqvrel 37573
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-qs 8711  df-coss 37794  df-coels 37795  df-refrel 37895  df-symrel 37927  df-trrel 37957  df-eqvrel 37968
This theorem is referenced by:  cpet2  38220
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