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Theorem fnmpt 5050
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
Hypothesis
Ref Expression
mptfng.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fnmpt  |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)    V( x)

Proof of Theorem fnmpt
StepHypRef Expression
1 elex 2611 . . 3  |-  ( B  e.  V  ->  B  e.  _V )
21ralimi 2427 . 2  |-  ( A. x  e.  A  B  e.  V  ->  A. x  e.  A  B  e.  _V )
3 mptfng.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
43mptfng 5049 . 2  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
52, 4sylib 120 1  |-  ( A. x  e.  A  B  e.  V  ->  F  Fn  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   A.wral 2349   _Vcvv 2602    |-> cmpt 3841    Fn wfn 4921
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-fun 4928  df-fn 4929
This theorem is referenced by:  mpt0  5051  ralrnmpt  5335  rexrnmpt  5336  fmpt  5345  fmpt2d  5353  f1ocnvd  5727  offval2  5751  ofrfval2  5752  caofinvl  5758  f1od2  5881  frectfr  6043  omfnex  6087  oeiv  6094
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