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Theorem nfunsn 5338
Description: If the restriction of a class to a singleton is not a function, its value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
nfunsn  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )

Proof of Theorem nfunsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eumo 1980 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4719 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 velsn 3463 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 3848 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 119 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 292 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 119 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2013 . . . . . . 7  |-  ( E* y  A F y  ->  E* y  x ( F  |`  { A } ) y )
101, 9syl 14 . . . . . 6  |-  ( E! y  A F y  ->  E* y  x ( F  |`  { A } ) y )
1110alrimiv 1802 . . . . 5  |-  ( E! y  A F y  ->  A. x E* y  x ( F  |`  { A } ) y )
12 relres 4741 . . . . 5  |-  Rel  ( F  |`  { A }
)
1311, 12jctil 305 . . . 4  |-  ( E! y  A F y  ->  ( Rel  ( F  |`  { A }
)  /\  A. x E* y  x ( F  |`  { A }
) y ) )
14 dffun6 5029 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1513, 14sylibr 132 . . 3  |-  ( E! y  A F y  ->  Fun  ( F  |` 
{ A } ) )
1615con3i 597 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  E! y  A F y )
17 tz6.12-2 5296 . 2  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1816, 17syl 14 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287    = wceq 1289    e. wcel 1438   E!weu 1948   E*wmo 1949   (/)c0 3286   {csn 3446   class class class wbr 3845    |` cres 4440   Rel wrel 4443   Fun wfun 5009   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-res 4450  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by: (None)
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