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Theorem mptfng 5573
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 3108 . . 3  |-  ( B  e.  _V  <->  E! y 
y  =  B )
21ralbii 2731 . 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 4271 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtri 2458 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
65fnopabg 5571 . 2  |-  ( A. x  e.  A  E! y  y  =  B  <->  F  Fn  A )
72, 6bitri 242 1  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2283   A.wral 2707   _Vcvv 2958   {copab 4268    e. cmpt 4269    Fn wfn 5452
This theorem is referenced by:  fnmpt  5574  fnmpti  5576  mpteqb  5822  bdayfo  25635  fobigcup  25750  ofmpteq  26789  dihf11lem  32137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-fun 5459  df-fn 5460
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