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Theorem cossex 37800
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 37795 . 2 𝐴 = (𝐴𝐴)
2 cnvexg 7911 . . 3 (𝐴𝑉𝐴 ∈ V)
3 coexg 7916 . . 3 ((𝐴𝑉𝐴 ∈ V) → (𝐴𝐴) ∈ V)
42, 3mpdan 684 . 2 (𝐴𝑉 → (𝐴𝐴) ∈ V)
51, 4eqeltrid 2831 1 (𝐴𝑉 → ≀ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3468  ccnv 5668  ccom 5673  ccoss 37554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-coss 37792
This theorem is referenced by:  cosscnvex  37801  1cosscnvepresex  37802  1cossxrncnvepresex  37803  cosselrels  37877  elfunsALTVfunALTV  38078  partimeq  38190
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