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Theorem mptfng 5533
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 3070 . . 3  |-  ( B  e.  _V  <->  E! y 
y  =  B )
21ralbii 2694 . 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 4232 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtri 2428 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
65fnopabg 5531 . 2  |-  ( A. x  e.  A  E! y  y  =  B  <->  F  Fn  A )
72, 6bitri 241 1  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E!weu 2258   A.wral 2670   _Vcvv 2920   {copab 4229    e. cmpt 4230    Fn wfn 5412
This theorem is referenced by:  fnmpt  5534  fnmpti  5536  mpteqb  5782  bdayfo  25547  fobigcup  25658  ofmpteq  26670  dihf11lem  31753
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-fun 5419  df-fn 5420
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