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Theorem coex 7952
Description: The composition of two sets is a set. (Contributed by NM, 15-Dec-2003.)
Hypotheses
Ref Expression
coex.1 𝐴 ∈ V
coex.2 𝐵 ∈ V
Assertion
Ref Expression
coex (𝐴𝐵) ∈ V

Proof of Theorem coex
StepHypRef Expression
1 coex.1 . 2 𝐴 ∈ V
2 coex.2 . 2 𝐵 ∈ V
3 coexg 7951 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3mp2an 692 1 (𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  Vcvv 3477  ccom 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699
This theorem is referenced by:  domtr  9045  enfixsn  9119  wdomtr  9612  cfcoflem  10309  axcc3  10475  axdc4uzlem  14020  hashfacen  14489  cofu1st  17933  cofu2nd  17935  cofucl  17938  fucid  18027  sursubmefmnd  18921  injsubmefmnd  18922  smndex1mgm  18932  gsumzaddlem  19953  cnfldfun  21395  cnfldfunALT  21396  cnfldfunOLD  21408  cnfldfunALTOLD  21409  cnfldfunALTOLDOLD  21410  znle  21568  selvval  22156  evls1fval  22338  evls1val  22339  evl1fval  22347  evl1val  22348  xkococnlem  23682  xkococn  23683  efmndtmd  24124  pserulm  26479  imsval  30713  tocycf  33119  eulerpartgbij  34353  derangenlem  35155  subfacp1lem5  35168  poimirlem9  37615  poimirlem15  37621  poimirlem17  37623  poimirlem20  37626  mbfresfi  37652  tendopl2  40759  erngplus2  40786  erngplus2-rN  40794  dvaplusgv  40992  dvhvaddass  41079  dvhlveclem  41090  diblss  41152  diblsmopel  41153  dicvaddcl  41172  dicvscacl  41173  cdlemn7  41185  dihordlem7  41196  dihopelvalcpre  41230  xihopellsmN  41236  dihopellsm  41237  rabren3dioph  42802  fzisoeu  45250  stirlinglem14  46042  fundcmpsurinjpreimafv  47332  grimco  47817  gricushgr  47823  uspgrlim  47894  grlictr  47910
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