Theorem coex 7855

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 7854	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 ∘ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2111  Vcvv 3436   ∘ ccom 5615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622

This theorem is referenced by:  domtr  8924  enfixsn  8994  wdomtr  9456  cfcoflem  10158  axcc3  10324  axdc4uzlem  13885  hashfacen  14356  cofu1st  17785  cofu2nd  17787  cofucl  17790  fucid  17876  sursubmefmnd  18799  injsubmefmnd  18800  smndex1mgm  18810  gsumzaddlem  19828  cnfldfun  21300  cnfldfunALT  21301  cnfldfunOLD  21313  cnfldfunALTOLD  21314  znle  21468  selvval  22045  evls1fval  22229  evls1val  22230  evl1fval  22238  evl1val  22239  xkococnlem  23569  xkococn  23570  efmndtmd  24011  pserulm  26353  imsval  30657  tocycf  33078  eulerpartgbij  34377  derangenlem  35207  subfacp1lem5  35220  poimirlem9  37669  poimirlem15  37675  poimirlem17  37677  poimirlem20  37680  mbfresfi  37706  tendopl2  40816  erngplus2  40843  erngplus2-rN  40851  dvaplusgv  41049  dvhvaddass  41136  dvhlveclem  41147  diblss  41209  diblsmopel  41210  dicvaddcl  41229  dicvscacl  41230  cdlemn7  41242  dihordlem7  41253  dihopelvalcpre  41287  xihopellsmN  41293  dihopellsm  41294  rabren3dioph  42848  fzisoeu  45341  stirlinglem14  46125  fundcmpsurinjpreimafv  47439  grimco  47920  gricushgr  47948  cycldlenngric  47959  uspgrlim  48023  grlictr  48046  fuco22natlem  49377