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Theorem cossex 38401
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 38396 . 2 𝐴 = (𝐴𝐴)
2 cnvexg 7947 . . 3 (𝐴𝑉𝐴 ∈ V)
3 coexg 7952 . . 3 ((𝐴𝑉𝐴 ∈ V) → (𝐴𝐴) ∈ V)
42, 3mpdan 687 . 2 (𝐴𝑉 → (𝐴𝐴) ∈ V)
51, 4eqeltrid 2843 1 (𝐴𝑉 → ≀ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  ccnv 5688  ccom 5693  ccoss 38162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-coss 38393
This theorem is referenced by:  cosscnvex  38402  1cosscnvepresex  38403  1cossxrncnvepresex  38404  cosselrels  38478  elfunsALTVfunALTV  38679  partimeq  38791
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