Theorem dmqs1cosscnvepreseq 39416

Description: Two ways to express the equality of the domain quotient of the coelements on the class 𝐴 with the class 𝐴. (Contributed by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
dmqs1cosscnvepreseq	⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)

Proof of Theorem dmqs1cosscnvepreseq

Step	Hyp	Ref	Expression
1		df-coels 39171	. . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴)
2	1	dmqseqeq1i 39397	. 2 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴)
3		dmqscoelseq 39415	. 2 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
4	2, 3	bitr3i 280	1 ⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1570  ∪ cuni 4872   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663   / cqs 8689   ≀ ccoss 38852   ∼ ccoels 38853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-coss 39170  df-coels 39171

This theorem is referenced by:  eqvreldmqs  39429  eqvreldmqs2  39430  eldisjn0el  39578