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Theorem coex 7160
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 7159 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3mp2an 708 1 (𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154
This theorem is referenced by:  domtr  8050  enfixsn  8110  wdomtr  8521  cfcoflem  9132  axcc3  9298  axdc4uzlem  12822  hashfacen  13276  cofu1st  16590  cofu2nd  16592  cofucl  16595  fucid  16678  symgplusg  17855  gsumzaddlem  18367  evls1fval  19732  evls1val  19733  evl1fval  19740  evl1val  19741  cnfldfun  19806  cnfldfunALT  19807  znle  19932  xkococnlem  21510  xkococn  21511  symgtgp  21952  pserulm  24221  imsval  27668  eulerpartgbij  30562  derangenlem  31279  subfacp1lem5  31292  poimirlem9  33548  poimirlem15  33554  poimirlem17  33556  poimirlem20  33559  mbfresfi  33586  tendopl2  36382  erngplus2  36409  erngplus2-rN  36417  dvaplusgv  36615  dvhvaddass  36703  dvhlveclem  36714  diblss  36776  diblsmopel  36777  dicvaddcl  36796  dicvscacl  36797  cdlemn7  36809  dihordlem7  36820  dihopelvalcpre  36854  xihopellsmN  36860  dihopellsm  36861  rabren3dioph  37696  fzisoeu  39828  stirlinglem14  40622
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