Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfcomember2 Structured version   Visualization version   GIF version

Theorem dfcomember2 38674
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.)
Assertion
Ref Expression
dfcomember2 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))

Proof of Theorem dfcomember2
StepHypRef Expression
1 dfcomember 38673 . 2 ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴)
2 dferALTV2 38669 . 2 ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
31, 2bitri 275 1 ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 /𝐴) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  dom cdm 5685   / cqs 8744  ccoels 38183   EqvRel weqvrel 38199   ErALTV werALTV 38208   CoMembEr wcomember 38210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coels 38413  df-refrel 38513  df-symrel 38545  df-trrel 38575  df-eqvrel 38586  df-dmqs 38640  df-erALTV 38665  df-comember 38667
This theorem is referenced by:  dfcomember3  38675
  Copyright terms: Public domain W3C validator