Theorem eqvreldmqs 39958

Description: Two ways to express comember equivalence relation on its domain quotient. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)

Assertion

Ref	Expression
eqvreldmqs	⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem eqvreldmqs

Step	Hyp	Ref	Expression
1		df-coeleqvrel 39869	. . 3 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴))
2	1	bicomi 227	. 2 ⊢ ( EqvRel ≀ (◡ E ↾ 𝐴) ↔ CoElEqvRel 𝐴)
3		dmqs1cosscnvepreseq 39945	. 2 ⊢ ((dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
4	2, 3	anbi12i 640	1 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∪ cuni 4872   E cep 5557  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660   / cqs 8754   ≀ ccoss 39381   ∼ ccoels 39382   EqvRel weqvrel 39398   CoElEqvRel wcoeleqvrel 39400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ec 8757  df-qs 8761  df-coss 39699  df-coels 39700  df-coeleqvrel 39869

This theorem is used by:  mpet3  40148