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Theorem fnmpoi 6228
Description: Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.)
Hypotheses
Ref Expression
fmpo.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
fnmpoi.2  |-  C  e. 
_V
Assertion
Ref Expression
fnmpoi  |-  F  Fn  ( A  X.  B
)
Distinct variable groups:    x, A, y   
x, B, y
Allowed substitution hints:    C( x, y)    F( x, y)

Proof of Theorem fnmpoi
StepHypRef Expression
1 fnmpoi.2 . . 3  |-  C  e. 
_V
21rgen2w 2546 . 2  |-  A. x  e.  A  A. y  e.  B  C  e.  _V
3 fmpo.1 . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43fnmpo 6226 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  ->  F  Fn  ( A  X.  B
) )
52, 4ax-mp 5 1  |-  F  Fn  ( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1364    e. wcel 2160   A.wral 2468   _Vcvv 2752    X. cxp 4642    Fn wfn 5230    e. cmpo 5897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165
This theorem is referenced by:  dmmpo  6229  fnoa  6471  fnom  6474  fnoei  6476  fnmap  6680  fnpm  6681  restfn  12745
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