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Theorem mapprc 6985
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 3610 . . 3  |-  ( { f  |  f : A --> B }  =/=  (/)  <->  E. f  f : A --> B )
2 fdm 5558 . . . . 5  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2923 . . . . . 6  |-  f  e. 
_V
43dmex 5095 . . . . 5  |-  dom  f  e.  _V
52, 4syl6eqelr 2497 . . . 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 2614 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 2394    =/= wne 2571   _Vcvv 2920   (/)c0 3592   dom cdm 4841   -->wf 5413
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 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-cnv 4849  df-dm 4851  df-rn 4852  df-fn 5420  df-f 5421
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