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Theorem nfunsn 5239
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 1974 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2605 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4646 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 velsn 3423 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 3796 . . . . . . . . . . 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 2007 . . . . . . 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 1796 . . . . 5  |-  ( E! y  A F y  ->  A. x E* y  x ( F  |`  { A } ) y )
12 relres 4667 . . . . 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 4946 . . . 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 595 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  E! y  A F y )
17 tz6.12-2 5200 . 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 1283    = wceq 1285    e. wcel 1434   E!weu 1942   E*wmo 1943   (/)c0 3258   {csn 3406   class class class wbr 3793    |` cres 4373   Rel wrel 4376   Fun wfun 4926   ` cfv 4932
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-res 4383  df-iota 4897  df-fun 4934  df-fv 4940
This theorem is referenced by: (None)
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