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Theorem mptfng 5725
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 3222 . . 3  |-  ( B  e.  _V  <->  E! y 
y  =  B )
21ralbii 2831 . 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 4477 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtri 2484 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
65fnopabg 5723 . 2  |-  ( A. x  e.  A  E! y  y  =  B  <->  F  Fn  A )
72, 6bitri 257 1  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455    e. wcel 1898   E!weu 2310   A.wral 2749   _Vcvv 3057   {copab 4474    |-> cmpt 4475    Fn wfn 5596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-fun 5603  df-fn 5604
This theorem is referenced by:  fnmpt  5726  fnmpti  5728  mpteqb  5987  ofmpteq  6577  bdayfo  30613  fobigcup  30716  dihf11lem  34879
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