Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ecex2 Structured version   Visualization version   GIF version

Theorem ecex2 35466
Description: Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
Assertion
Ref Expression
ecex2 ((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))

Proof of Theorem ecex2
StepHypRef Expression
1 ecexg 8282 . 2 ((𝑅𝐴) ∈ 𝑉 → [𝐵](𝑅𝐴) ∈ V)
2 ecres2 35417 . . 3 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
32eleq1d 2894 . 2 (𝐵𝐴 → ([𝐵](𝑅𝐴) ∈ V ↔ [𝐵]𝑅 ∈ V))
41, 3syl5ibcom 246 1 ((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  Vcvv 3492  cres 5550  [cec 8276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ec 8280
This theorem is referenced by:  uniqsALTV  35467
  Copyright terms: Public domain W3C validator