Theorem coex 7870

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 7869	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 ∘ 𝐵) ∈ V)
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 ∘ 𝐵) ∈ V

Syntax hints:   ∈ wcel 2109  Vcvv 3438   ∘ ccom 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634

This theorem is referenced by:  domtr  8939  enfixsn  9010  wdomtr  9486  cfcoflem  10185  axcc3  10351  axdc4uzlem  13909  hashfacen  14380  cofu1st  17809  cofu2nd  17811  cofucl  17814  fucid  17900  sursubmefmnd  18789  injsubmefmnd  18790  smndex1mgm  18800  gsumzaddlem  19819  cnfldfun  21294  cnfldfunALT  21295  cnfldfunOLD  21307  cnfldfunALTOLD  21308  znle  21462  selvval  22039  evls1fval  22223  evls1val  22224  evl1fval  22232  evl1val  22233  xkococnlem  23563  xkococn  23564  efmndtmd  24005  pserulm  26348  imsval  30648  tocycf  33078  eulerpartgbij  34359  derangenlem  35163  subfacp1lem5  35176  poimirlem9  37628  poimirlem15  37634  poimirlem17  37636  poimirlem20  37639  mbfresfi  37665  tendopl2  40776  erngplus2  40803  erngplus2-rN  40811  dvaplusgv  41009  dvhvaddass  41096  dvhlveclem  41107  diblss  41169  diblsmopel  41170  dicvaddcl  41189  dicvscacl  41190  cdlemn7  41202  dihordlem7  41213  dihopelvalcpre  41247  xihopellsmN  41253  dihopellsm  41254  rabren3dioph  42808  fzisoeu  45302  stirlinglem14  46088  fundcmpsurinjpreimafv  47412  grimco  47893  gricushgr  47921  cycldlenngric  47932  uspgrlim  47996  grlictr  48019  fuco22natlem  49350