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Theorem mapprc 6771
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 3474 . . 3  |-  ( { f  |  f : A --> B }  =/=  (/)  <->  E. f  f : A --> B )
2 fdm 5358 . . . . 5  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2792 . . . . . 6  |-  f  e. 
_V
43dmex 4940 . . . . 5  |-  dom  f  e.  _V
52, 4syl6eqelr 2373 . . . 4  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1670 . . 3  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 189 . 2  |-  ( { f  |  f : A --> B }  =/=  (/) 
->  A  e.  _V )
87necon1bi 2490 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1533    = wceq 1628    e. wcel 1688   {cab 2270    =/= wne 2447   _Vcvv 2789   (/)c0 3456   dom cdm 4688   -->wf 5217
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-cnv 4696  df-dm 4698  df-rn 4699  df-fn 5224  df-f 5225
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