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Theorem fnmpt2i 5830
Description: Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.)
Hypotheses
Ref Expression
fmpt2.1  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
fnmpt2i.2  |-  C  e. 
_V
Assertion
Ref Expression
fnmpt2i  |-  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 fnmpt2i
StepHypRef Expression
1 fnmpt2i.2 . . 3  |-  C  e. 
_V
21rgen2w 2377 . 2  |-  A. x  e.  A  A. y  e.  B  C  e.  _V
3 fmpt2.1 . . 3  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
43fnmpt2 5828 . 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 1243    e. wcel 1393   A.wral 2306   _Vcvv 2557    X. cxp 4343    Fn wfn 4897    |-> cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768
This theorem is referenced by:  dmmpt2  5831  fnoa  6027  fnom  6030  fnoei  6032
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