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Theorem coex 7235
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 7234 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
41, 2, 3mp2an 710 1 (𝐴𝐵) ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2103  Vcvv 3304  ccom 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229
This theorem is referenced by:  domtr  8125  enfixsn  8185  wdomtr  8596  cfcoflem  9207  axcc3  9373  axdc4uzlem  12897  hashfacen  13351  cofu1st  16665  cofu2nd  16667  cofucl  16670  fucid  16753  symgplusg  17930  gsumzaddlem  18442  evls1fval  19807  evls1val  19808  evl1fval  19815  evl1val  19816  cnfldfun  19881  cnfldfunALT  19882  znle  20007  xkococnlem  21585  xkococn  21586  symgtgp  22027  pserulm  24296  imsval  27770  eulerpartgbij  30664  derangenlem  31381  subfacp1lem5  31394  poimirlem9  33650  poimirlem15  33656  poimirlem17  33658  poimirlem20  33661  mbfresfi  33688  tendopl2  36484  erngplus2  36511  erngplus2-rN  36519  dvaplusgv  36717  dvhvaddass  36805  dvhlveclem  36816  diblss  36878  diblsmopel  36879  dicvaddcl  36898  dicvscacl  36899  cdlemn7  36911  dihordlem7  36922  dihopelvalcpre  36956  xihopellsmN  36962  dihopellsm  36963  rabren3dioph  37798  fzisoeu  39930  stirlinglem14  40724
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