Theorem dfcomember3 38201

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 38200	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfcoeleqvrel 38149	. . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
3	2	bicomi 223	. . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4		dmqscoelseq 38188	. . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	3, 4	anbi12i 626	. 2 ⊢ (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
6	1, 5	bitri 274	1 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  ∪ cuni 4903  dom cdm 5672   / cqs 8720   ∼ ccoels 37705   EqvRel weqvrel 37721   CoElEqvRel wcoeleqvrel 37723   CoMembEr wcomember 37732

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  df-coeleqvrel 38114  df-dmqs 38166  df-erALTV 38191  df-comember 38193

This theorem is referenced by:  mainer  38361  mpet  38366