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Theorem nfunsn 5462
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 2032 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2692 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4832 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 velsn 3548 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 3939 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 120 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 296 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 120 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2065 . . . . . . 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 1847 . . . . 5  |-  ( E! y  A F y  ->  A. x E* y  x ( F  |`  { A } ) y )
12 relres 4854 . . . . 5  |-  Rel  ( F  |`  { A }
)
1311, 12jctil 310 . . . 4  |-  ( E! y  A F y  ->  ( Rel  ( F  |`  { A }
)  /\  A. x E* y  x ( F  |`  { A }
) y ) )
14 dffun6 5144 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1513, 14sylibr 133 . . 3  |-  ( E! y  A F y  ->  Fun  ( F  |` 
{ A } ) )
1615con3i 622 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  E! y  A F y )
17 tz6.12-2 5419 . 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 103    <-> wb 104   A.wal 1330    = wceq 1332    e. wcel 1481   E!weu 2000   E*wmo 2001   (/)c0 3367   {csn 3531   class class class wbr 3936    |` cres 4548   Rel wrel 4551   Fun wfun 5124   ` cfv 5130
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-res 4558  df-iota 5095  df-fun 5132  df-fv 5138
This theorem is referenced by: (None)
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