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Theorem fnmpt2 6419
Description: Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypothesis
Ref Expression
fmpt2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
fnmpt2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
Distinct variable groups:    x, A, y    x, B, y
Allowed substitution hints:    C( x, y)    F( x, y)    V( x, y)

Proof of Theorem fnmpt2
StepHypRef Expression
1 elex 2964 . . . 4  |-  ( C  e.  V  ->  C  e.  _V )
21ralimi 2781 . . 3  |-  ( A. y  e.  B  C  e.  V  ->  A. y  e.  B  C  e.  _V )
32ralimi 2781 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  A. x  e.  A  A. y  e.  B  C  e.  _V )
4 fmpt2.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
54fmpt2 6418 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  <->  F : ( A  X.  B ) --> _V )
6 dffn2 5592 . . 3  |-  ( F  Fn  ( A  X.  B )  <->  F :
( A  X.  B
) --> _V )
75, 6bitr4i 244 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  e.  _V  <->  F  Fn  ( A  X.  B ) )
83, 7sylib 189 1  |-  ( A. x  e.  A  A. y  e.  B  C  e.  V  ->  F  Fn  ( A  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    X. cxp 4876    Fn wfn 5449   -->wf 5450    e. cmpt2 6083
This theorem is referenced by:  fnmpt2i  6420  genpdm  8879  mpt2cti  24110  pstmxmet  24292  cnre2csqima  24309  dmmpt2g  27959
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350
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