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Theorem mulex 10400
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 8862 . 2  |-  x.  :
( CC  X.  CC )
--> CC
2 cnex 8863 . . 3  |-  CC  e.  _V
32, 2xpex 4838 . 2  |-  ( CC 
X.  CC )  e. 
_V
4 fex2 5439 . 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 1701   _Vcvv 2822    X. cxp 4724   -->wf 5288   CCcc 8780    x. cmul 8787
This theorem is referenced by:  cnfldmul  16438  cnrngo  21123  cnnvg  21301  cnnvs  21304  cncph  21452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-xp 4732  df-rel 4733  df-cnv 4734  df-dm 4736  df-rn 4737  df-fun 5294  df-fn 5295  df-f 5296
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