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Theorem coex 7634
 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 7633 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3mp2an 690 1 (𝐴𝐵) ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∈ wcel 2110  Vcvv 3494   ∘ ccom 5558 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565 This theorem is referenced by:  domtr  8561  enfixsn  8625  wdomtr  9038  cfcoflem  9693  axcc3  9859  axdc4uzlem  13350  hashfacen  13811  cofu1st  17152  cofu2nd  17154  cofucl  17157  fucid  17240  sursubmefmnd  18060  injsubmefmnd  18061  smndex1mgm  18071  gsumzaddlem  19040  selvval  20330  evls1fval  20481  evls1val  20482  evl1fval  20490  evl1val  20491  cnfldfun  20556  cnfldfunALT  20557  znle  20682  xkococnlem  22266  xkococn  22267  efmndtmd  22708  pserulm  25009  imsval  28461  tocycf  30759  eulerpartgbij  31630  derangenlem  32418  subfacp1lem5  32431  poimirlem9  34900  poimirlem15  34906  poimirlem17  34908  poimirlem20  34911  mbfresfi  34937  tendopl2  37912  erngplus2  37939  erngplus2-rN  37947  dvaplusgv  38145  dvhvaddass  38232  dvhlveclem  38243  diblss  38305  diblsmopel  38306  dicvaddcl  38325  dicvscacl  38326  cdlemn7  38338  dihordlem7  38349  dihopelvalcpre  38383  xihopellsmN  38389  dihopellsm  38390  rabren3dioph  39412  fzisoeu  41567  stirlinglem14  42373  fundcmpsurinjpreimafv  43569  isomushgr  43992  isomgrtr  44005
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