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Theorem mapprc 6423
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 3312 . . . 4  |-  ( E. g  g  e.  {
f  |  f : A --> B }  <->  E. f 
f : A --> B )
2 fdm 5179 . . . . . 6  |-  ( f : A --> B  ->  dom  f  =  A
)
3 vex 2623 . . . . . . 7  |-  f  e. 
_V
43dmex 4712 . . . . . 6  |-  dom  f  e.  _V
52, 4syl6eqelr 2180 . . . . 5  |-  ( f : A --> B  ->  A  e.  _V )
65exlimiv 1535 . . . 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 598 . 2  |-  ( -.  A  e.  _V  ->  -. 
E. g  g  e. 
{ f  |  f : A --> B }
)
9 notm0 3307 . 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 1290   E.wex 1427    e. wcel 1439   {cab 2075   _Vcvv 2620   (/)c0 3287   dom cdm 4451   -->wf 5024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-rex 2366  df-v 2622  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-cnv 4459  df-dm 4461  df-rn 4462  df-fn 5031  df-f 5032
This theorem is referenced by: (None)
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