Theorem coex 7970

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 7969	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	mp2an 691	1 ⊢ (𝐴 ∘ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2108  Vcvv 3488   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711

This theorem is referenced by:  domtr  9067  enfixsn  9147  wdomtr  9644  cfcoflem  10341  axcc3  10507  axdc4uzlem  14034  hashfacen  14503  cofu1st  17947  cofu2nd  17949  cofucl  17952  fucid  18041  sursubmefmnd  18931  injsubmefmnd  18932  smndex1mgm  18942  gsumzaddlem  19963  cnfldfun  21401  cnfldfunALT  21402  cnfldfunOLD  21414  cnfldfunALTOLD  21415  cnfldfunALTOLDOLD  21416  znle  21574  selvval  22162  evls1fval  22344  evls1val  22345  evl1fval  22353  evl1val  22354  xkococnlem  23688  xkococn  23689  efmndtmd  24130  pserulm  26483  imsval  30717  tocycf  33110  eulerpartgbij  34337  derangenlem  35139  subfacp1lem5  35152  poimirlem9  37589  poimirlem15  37595  poimirlem17  37597  poimirlem20  37600  mbfresfi  37626  tendopl2  40734  erngplus2  40761  erngplus2-rN  40769  dvaplusgv  40967  dvhvaddass  41054  dvhlveclem  41065  diblss  41127  diblsmopel  41128  dicvaddcl  41147  dicvscacl  41148  cdlemn7  41160  dihordlem7  41171  dihopelvalcpre  41205  xihopellsmN  41211  dihopellsm  41212  rabren3dioph  42771  fzisoeu  45215  stirlinglem14  46008  fundcmpsurinjpreimafv  47282  grimco  47764  gricushgr  47770  uspgrlim  47816  grlictr  47832