Theorem dfcomember3 39206

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 39205	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfcoeleqvrel 39153	. . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
3	2	bicomi 226	. . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4		dmqscoelseq 39193	. . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	3, 4	anbi12i 636	. 2 ⊢ (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
6	1, 5	bitri 277	1 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1554  ∪ cuni 4859  dom cdm 5640   / cqs 8665   ∼ ccoels 38631   EqvRel weqvrel 38647   CoElEqvRel wcoeleqvrel 38649   CoMembEr wcomember 38660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-opab 5157  df-eprel 5540  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-ec 8668  df-qs 8672  df-coss 38948  df-coels 38949  df-refrel 39039  df-symrel 39071  df-trrel 39105  df-eqvrel 39116  df-coeleqvrel 39118  df-dmqs 39170  df-erALTV 39196  df-comember 39198

This theorem is referenced by:  mainer  39395  mpet  39400