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Theorem dfcomember3 37482
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
StepHypRef Expression
1 dfcomember2 37481 . 2 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
2 dfcoeleqvrel 37430 . . . 4 ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴)
32bicomi 223 . . 3 ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴)
4 dmqscoelseq 37469 . . 3 ((dom ∼ 𝐴 /𝐴) = 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴)
53, 4anbi12i 628 . 2 (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
61, 5bitri 275 1 ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ ( 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542   cuni 4907  dom cdm 5675   / cqs 8698  ccoels 36982   EqvRel weqvrel 36998   CoElEqvRel wcoeleqvrel 37000   CoMembEr wcomember 37009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8701  df-qs 8705  df-coss 37219  df-coels 37220  df-refrel 37320  df-symrel 37352  df-trrel 37382  df-eqvrel 37393  df-coeleqvrel 37395  df-dmqs 37447  df-erALTV 37472  df-comember 37474
This theorem is referenced by:  mainer  37642  mpet  37647
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