Theorem dfcomember3 38931

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 38930	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfcoeleqvrel 38878	. . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
3	2	bicomi 224	. . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4		dmqscoelseq 38918	. . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	3, 4	anbi12i 629	. 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 1542  ∪ cuni 4864  dom cdm 5625   / cqs 8636   ∼ ccoels 38356   EqvRel weqvrel 38372   CoElEqvRel wcoeleqvrel 38374   CoMembEr wcomember 38385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-qs 8643  df-coss 38673  df-coels 38674  df-refrel 38764  df-symrel 38796  df-trrel 38830  df-eqvrel 38841  df-coeleqvrel 38843  df-dmqs 38895  df-erALTV 38921  df-comember 38923

This theorem is referenced by:  mainer  39120  mpet  39125