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

Theorem cossex 36870
Description: If 𝐴 is a set then the class of cosets by 𝐴 is a set. (Contributed by Peter Mazsa, 4-Jan-2019.)
Assertion
Ref Expression
cossex (𝐴𝑉 → ≀ 𝐴 ∈ V)

Proof of Theorem cossex
StepHypRef Expression
1 dfcoss3 36865 . 2 𝐴 = (𝐴𝐴)
2 cnvexg 7858 . . 3 (𝐴𝑉𝐴 ∈ V)
3 coexg 7863 . . 3 ((𝐴𝑉𝐴 ∈ V) → (𝐴𝐴) ∈ V)
42, 3mpdan 685 . 2 (𝐴𝑉 → (𝐴𝐴) ∈ V)
51, 4eqeltrid 2842 1 (𝐴𝑉 → ≀ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3444  ccnv 5631  ccom 5636  ccoss 36623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-coss 36862
This theorem is referenced by:  cosscnvex  36871  1cosscnvepresex  36872  1cossxrncnvepresex  36873  cosselrels  36947  elfunsALTVfunALTV  37148  partimeq  37260
  Copyright terms: Public domain W3C validator