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Theorem mapprc 6981
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 3606 . . 3  |-  ( { f  |  f : A --> B }  =/=  (/)  <->  E. f  f : A --> B )
2 fdm 5554 . . . . 5  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2919 . . . . . 6  |-  f  e. 
_V
43dmex 5091 . . . . 5  |-  dom  f  e.  _V
52, 4syl6eqelr 2493 . . . 4  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1641 . . 3  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 188 . 2  |-  ( { f  |  f : A --> B }  =/=  (/) 
->  A  e.  _V )
87necon1bi 2610 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   _Vcvv 2916   (/)c0 3588   dom cdm 4837   -->wf 5409
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-cnv 4845  df-dm 4847  df-rn 4848  df-fn 5416  df-f 5417
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