MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapprc Unicode version

Theorem mapprc 6662
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 3380 . . 3  |-  ( { f  |  f : A --> B }  =/=  (/)  <->  E. f  f : A --> B )
2 fdm 5250 . . . . 5  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2730 . . . . . 6  |-  f  e. 
_V
43dmex 4848 . . . . 5  |-  dom  f  e.  _V
52, 4syl6eqelr 2342 . . . 4  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 2023 . . 3  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 189 . 2  |-  ( { f  |  f : A --> B }  =/=  (/) 
->  A  e.  _V )
87necon1bi 2455 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2239    =/= wne 2412   _Vcvv 2727   (/)c0 3362   dom cdm 4580   -->wf 4588
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-cnv 4596  df-dm 4598  df-rn 4599  df-fn 4603  df-f 4604
  Copyright terms: Public domain W3C validator