Theorem dfcomember3 38666

Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.)

Assertion

Ref	Expression
dfcomember3	⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem dfcomember3

Step	Hyp	Ref	Expression
1		dfcomember2 38665	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfcoeleqvrel 38613	. . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
3	2	bicomi 224	. . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4		dmqscoelseq 38653	. . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	3, 4	anbi12i 628	. 2 ⊢ (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
6	1, 5	bitri 275	1 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∪ cuni 4871  dom cdm 5638   / cqs 8670   ∼ ccoels 38170   EqvRel weqvrel 38186   CoElEqvRel wcoeleqvrel 38188   CoMembEr wcomember 38197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677  df-coss 38402  df-coels 38403  df-refrel 38503  df-symrel 38535  df-trrel 38565  df-eqvrel 38576  df-coeleqvrel 38578  df-dmqs 38630  df-erALTV 38656  df-comember 38658

This theorem is referenced by:  mainer  38826  mpet  38831