Theorem coex 7874

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	⊢ (𝐴 ∘ 𝐵) ∈ V

Proof of Theorem coex

Step	Hyp	Ref	Expression
1		coex.1	. 2 ⊢ 𝐴 ∈ V
2		coex.2	. 2 ⊢ 𝐵 ∈ V
3		coexg 7873	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	mp2an 693	1 ⊢ (𝐴 ∘ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2114  Vcvv 3430   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635

This theorem is referenced by:  domtr  8947  enfixsn  9017  wdomtr  9483  cfcoflem  10185  axcc3  10351  axdc4uzlem  13936  hashfacen  14407  cofu1st  17841  cofu2nd  17843  cofucl  17846  fucid  17932  sursubmefmnd  18855  injsubmefmnd  18856  smndex1mgm  18869  gsumzaddlem  19887  cnfldfun  21358  cnfldfunALT  21359  cnfldfunOLD  21371  cnfldfunALTOLD  21372  znle  21526  selvval  22111  evls1fval  22294  evls1val  22295  evl1fval  22303  evl1val  22304  xkococnlem  23634  xkococn  23635  efmndtmd  24076  pserulm  26400  imsval  30771  tocycf  33193  eulerpartgbij  34532  derangenlem  35369  subfacp1lem5  35382  poimirlem9  37964  poimirlem15  37970  poimirlem17  37972  poimirlem20  37975  mbfresfi  38001  tendopl2  41237  erngplus2  41264  erngplus2-rN  41272  dvaplusgv  41470  dvhvaddass  41557  dvhlveclem  41568  diblss  41630  diblsmopel  41631  dicvaddcl  41650  dicvscacl  41651  cdlemn7  41663  dihordlem7  41674  dihopelvalcpre  41708  xihopellsmN  41714  dihopellsm  41715  rabren3dioph  43261  fzisoeu  45751  stirlinglem14  46533  fundcmpsurinjpreimafv  47880  grimco  48377  gricushgr  48405  cycldlenngric  48416  uspgrlim  48480  grlictr  48503  fuco22natlem  49832