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Theorem mapprc 6666
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 3460 . . . 4  |-  ( E. g  g  e.  {
f  |  f : A --> B }  <->  E. f 
f : A --> B )
2 fdm 5383 . . . . . 6  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2752 . . . . . . 7  |-  f  e. 
_V
43dmex 4905 . . . . . 6  |-  dom  f  e.  _V
52, 4eqeltrrdi 2279 . . . . 5  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1608 . . . 4  |-  ( E. f  f : A --> B  ->  A  e.  _V )
71, 6sylbi 121 . . 3  |-  ( E. g  g  e.  {
f  |  f : A --> B }  ->  A  e.  _V )
87con3i 633 . 2  |-  ( -.  A  e.  _V  ->  -. 
E. g  g  e. 
{ f  |  f : A --> B }
)
9 notm0 3455 . 2  |-  ( -. 
E. g  g  e. 
{ f  |  f : A --> B }  <->  { f  |  f : A --> B }  =  (/) )
108, 9sylib 122 1  |-  ( -.  A  e.  _V  ->  { f  |  f : A --> B }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1363   E.wex 1502    e. wcel 2158   {cab 2173   _Vcvv 2749   (/)c0 3434   dom cdm 4638   -->wf 5224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649  df-fn 5231  df-f 5232
This theorem is referenced by: (None)
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