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Theorem mptfng 5308
Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
Hypothesis
Ref Expression
mptfng.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptfng  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem mptfng
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eueq 2893 . . 3  |-  ( B  e.  _V  <->  E! y 
y  =  B )
21ralbii 2470 . 2  |-  ( A. x  e.  A  B  e.  _V  <->  A. x  e.  A  E! y  y  =  B )
3 mptfng.1 . . . 4  |-  F  =  ( x  e.  A  |->  B )
4 df-mpt 4040 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtri 2185 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
65fnopabg 5306 . 2  |-  ( A. x  e.  A  E! y  y  =  B  <->  F  Fn  A )
72, 6bitri 183 1  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1342   E!weu 2013    e. wcel 2135   A.wral 2442   _Vcvv 2722   {copab 4037    |-> cmpt 4038    Fn wfn 5178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-mpt 4040  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-fun 5185  df-fn 5186
This theorem is referenced by:  fnmpt  5309  fnmpti  5311  mpteqb  5571  cc3  7201
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