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Theorem mapprc 6626
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
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 abn0m 3439 . . . 4  |-  ( E. g  g  e.  {
f  |  f : A --> B }  <->  E. f 
f : A --> B )
2 fdm 5351 . . . . . 6  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2733 . . . . . . 7  |-  f  e. 
_V
43dmex 4875 . . . . . 6  |-  dom  f  e.  _V
52, 4eqeltrrdi 2262 . . . . 5  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1591 . . . 4  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 120 . . 3  |-  ( E. g  g  e.  {
f  |  f : A --> B }  ->  A  e.  _V )
87con3i 627 . 2  |-  ( -.  A  e.  _V  ->  -. 
E. g  g  e. 
{ f  |  f : A --> B }
)
9 notm0 3434 . 2  |-  ( -. 
E. g  g  e. 
{ f  |  f : A --> B }  <->  { f  |  f : A --> B }  =  (/) )
108, 9sylib 121 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1348   E.wex 1485    e. wcel 2141   {cab 2156   _Vcvv 2730   (/)c0 3414   dom cdm 4609   -->wf 5192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-cnv 4617  df-dm 4619  df-rn 4620  df-fn 5199  df-f 5200
This theorem is referenced by: (None)
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