MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coex Structured version   Visualization version   GIF version

Theorem coex 7921
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 7920 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3mp2an 691 1 (𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3475  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688
This theorem is referenced by:  domtr  9003  enfixsn  9081  wdomtr  9570  cfcoflem  10267  axcc3  10433  axdc4uzlem  13948  hashfacen  14413  hashfacenOLD  14414  cofu1st  17833  cofu2nd  17835  cofucl  17838  fucid  17924  sursubmefmnd  18777  injsubmefmnd  18778  smndex1mgm  18788  gsumzaddlem  19789  cnfldfun  20956  cnfldfunALT  20957  cnfldfunALTOLD  20958  znle  21088  selvval  21681  evls1fval  21838  evls1val  21839  evl1fval  21847  evl1val  21848  xkococnlem  23163  xkococn  23164  efmndtmd  23605  pserulm  25934  imsval  29938  tocycf  32276  eulerpartgbij  33371  derangenlem  34162  subfacp1lem5  34175  poimirlem9  36497  poimirlem15  36503  poimirlem17  36505  poimirlem20  36508  mbfresfi  36534  tendopl2  39648  erngplus2  39675  erngplus2-rN  39683  dvaplusgv  39881  dvhvaddass  39968  dvhlveclem  39979  diblss  40041  diblsmopel  40042  dicvaddcl  40061  dicvscacl  40062  cdlemn7  40074  dihordlem7  40085  dihopelvalcpre  40119  xihopellsmN  40125  dihopellsm  40126  rabren3dioph  41553  fzisoeu  44010  stirlinglem14  44803  fundcmpsurinjpreimafv  46076  isomushgr  46494  isomgrtr  46507
  Copyright terms: Public domain W3C validator