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Theorem fnmpoi 6056
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 2462 . 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 6054 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  ->  F  Fn  ( A  X.  B
) )
52, 4ax-mp 7 1  |-  F  Fn  ( A  X.  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1314    e. wcel 1463   A.wral 2390   _Vcvv 2657    X. cxp 4497    Fn wfn 5076    e. cmpo 5730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993
This theorem is referenced by:  dmmpo  6057  fnoa  6297  fnom  6300  fnoei  6302  fnmap  6503  fnpm  6504  restfn  11967
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