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Theorem mapprc 6864
Description: When  A is a proper class, the class of all functions mapping  A to  B is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.)
Assertion
Ref Expression
mapprc  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Distinct variable groups:    A, f    B, f

Proof of Theorem mapprc
StepHypRef Expression
1 abn0 3549 . . 3  |-  ( { f  |  f : A --> B }  =/=  (/)  <->  E. f  f : A --> B )
2 fdm 5476 . . . . 5  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2867 . . . . . 6  |-  f  e. 
_V
43dmex 5023 . . . . 5  |-  dom  f  e.  _V
52, 4syl6eqelr 2447 . . . 4  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1634 . . 3  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 187 . 2  |-  ( { f  |  f : A --> B }  =/=  (/) 
->  A  e.  _V )
87necon1bi 2564 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   _Vcvv 2864   (/)c0 3531   dom cdm 4771   -->wf 5333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-cnv 4779  df-dm 4781  df-rn 4782  df-fn 5340  df-f 5341
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