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Theorem nfunsn 5589
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 2074 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2763 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4948 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 velsn 3635 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4032 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 121 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 298 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 121 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2107 . . . . . . 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 1885 . . . . 5  |-  ( E! y  A F y  ->  A. x E* y  x ( F  |`  { A } ) y )
12 relres 4970 . . . . 5  |-  Rel  ( F  |`  { A }
)
1311, 12jctil 312 . . . 4  |-  ( E! y  A F y  ->  ( Rel  ( F  |`  { A }
)  /\  A. x E* y  x ( F  |`  { A }
) y ) )
14 dffun6 5268 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1513, 14sylibr 134 . . 3  |-  ( E! y  A F y  ->  Fun  ( F  |` 
{ A } ) )
1615con3i 633 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  E! y  A F y )
17 tz6.12-2 5545 . 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 104    <-> wb 105   A.wal 1362    = wceq 1364   E!weu 2042   E*wmo 2043    e. wcel 2164   (/)c0 3446   {csn 3618   class class class wbr 4029    |` cres 4661   Rel wrel 4664   Fun wfun 5248   ` cfv 5254
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-res 4671  df-iota 5215  df-fun 5256  df-fv 5262
This theorem is referenced by: (None)
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