Theorem dfcomember3 39270

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 39269	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfcoeleqvrel 39217	. . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
3	2	bicomi 227	. . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4		dmqscoelseq 39257	. . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)
5	3, 4	anbi12i 639	. 2 ⊢ (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
6	1, 5	bitri 278	1 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1563  ∪ cuni 4868  dom cdm 5652   / cqs 8681   ∼ ccoels 38695   EqvRel weqvrel 38711   CoElEqvRel wcoeleqvrel 38713   CoMembEr wcomember 38724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-eprel 5552  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ec 8684  df-qs 8688  df-coss 39012  df-coels 39013  df-refrel 39103  df-symrel 39135  df-trrel 39169  df-eqvrel 39180  df-coeleqvrel 39182  df-dmqs 39234  df-erALTV 39260  df-comember 39262

This theorem is referenced by:  mainer  39459  mpet  39464