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Theorem mulex 10353
Description: The multiplication operation is a set. (Contributed by NM, 19-Oct-2004.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
mulex  |-  x.  e.  _V

Proof of Theorem mulex
StepHypRef Expression
1 ax-mulf 8817 . 2  |-  x.  :
( CC  X.  CC )
--> CC
2 cnex 8818 . . 3  |-  CC  e.  _V
32, 2xpex 4801 . 2  |-  ( CC 
X.  CC )  e. 
_V
4 fex2 5401 . 2  |-  ( (  x.  : ( CC 
X.  CC ) --> CC 
/\  ( CC  X.  CC )  e.  _V  /\  CC  e.  _V )  ->  x.  e.  _V )
51, 3, 2, 4mp3an 1277 1  |-  x.  e.  _V
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   _Vcvv 2788    X. cxp 4687   -->wf 5251   CCcc 8735    x. cmul 8742
This theorem is referenced by:  cnfldmul  16385  cnrngo  21070  cnnvg  21246  cnnvs  21249  cncph  21397  zintdom  25438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259
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