Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eqvreldisj4 Structured version   Visualization version   GIF version

Theorem eqvreldisj4 38806
Description: Intersection with the converse epsilon relation restricted to the quotient set of an equivalence relation is disjoint. (Contributed by Peter Mazsa, 30-May-2020.) (Revised by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
eqvreldisj4 ( EqvRel 𝑅 → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))

Proof of Theorem eqvreldisj4
StepHypRef Expression
1 eqvreldisj3 38805 . 2 ( EqvRel 𝑅 → Disj ( E ↾ (𝐵 / 𝑅)))
2 disjimin 38730 . 2 ( Disj ( E ↾ (𝐵 / 𝑅)) → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))
31, 2syl 17 1 ( EqvRel 𝑅 → Disj (𝑆 ∩ ( E ↾ (𝐵 / 𝑅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3949   E cep 5581  ccnv 5682  cres 5685   / cqs 8740   EqvRel weqvrel 38177   Disj wdisjALTV 38194
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
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-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-eprel 5582  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-ec 8743  df-qs 8747  df-coss 38390  df-refrel 38491  df-cnvrefrel 38506  df-symrel 38523  df-trrel 38553  df-eqvrel 38564  df-funALTV 38661  df-disjALTV 38684  df-eldisj 38686
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator