ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mptfng Unicode version

Theorem mptfng 5139
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 2786 . . 3  |-  ( B  e.  _V  <->  E! y 
y  =  B )
21ralbii 2384 . 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 3901 . . . 4  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
53, 4eqtri 2108 . . 3  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
65fnopabg 5137 . 2  |-  ( A. x  e.  A  E! y  y  =  B  <->  F  Fn  A )
72, 6bitri 182 1  |-  ( A. x  e.  A  B  e.  _V  <->  F  Fn  A
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E!weu 1948   A.wral 2359   _Vcvv 2619   {copab 3898    |-> cmpt 3899    Fn wfn 5010
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-fun 5017  df-fn 5018
This theorem is referenced by:  fnmpt  5140  fnmpti  5142  mpteqb  5393
  Copyright terms: Public domain W3C validator