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Theorem cossex 37891
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 37886 . 2 𝐴 = (𝐴𝐴)
2 cnvexg 7932 . . 3 (𝐴𝑉𝐴 ∈ V)
3 coexg 7937 . . 3 ((𝐴𝑉𝐴 ∈ V) → (𝐴𝐴) ∈ V)
42, 3mpdan 686 . 2 (𝐴𝑉 → (𝐴𝐴) ∈ V)
51, 4eqeltrid 2833 1 (𝐴𝑉 → ≀ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471  ccnv 5677  ccom 5682  ccoss 37648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-coss 37883
This theorem is referenced by:  cosscnvex  37892  1cosscnvepresex  37893  1cossxrncnvepresex  37894  cosselrels  37968  elfunsALTVfunALTV  38169  partimeq  38281
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