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Theorem nfunsn 5896
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 2328 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 3048 . . . . . . . . . 10  |-  y  e. 
_V
32brres 5111 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 elsn 3982 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4405 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 199 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 489 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 199 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2349 . . . . . . 7  |-  ( E* y  A F y  ->  E* y  x ( F  |`  { A } ) y )
101, 9syl 17 . . . . . 6  |-  ( E! y  A F y  ->  E* y  x ( F  |`  { A } ) y )
11 tz6.12-2 5856 . . . . . 6  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1210, 11nsyl4 148 . . . . 5  |-  ( -.  ( F `  A
)  =  (/)  ->  E* y  x ( F  |`  { A } ) y )
1312alrimiv 1773 . . . 4  |-  ( -.  ( F `  A
)  =  (/)  ->  A. x E* y  x ( F  |`  { A }
) y )
14 relres 5132 . . . 4  |-  Rel  ( F  |`  { A }
)
1513, 14jctil 540 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
16 dffun6 5597 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1715, 16sylibr 216 . 2  |-  ( -.  ( F `  A
)  =  (/)  ->  Fun  ( F  |`  { A } ) )
1817con1i 133 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444    e. wcel 1887   E!weu 2299   E*wmo 2300   (/)c0 3731   {csn 3968   class class class wbr 4402    |` cres 4836   Rel wrel 4839   Fun wfun 5576   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590
This theorem is referenced by:  fvfundmfvn0  5897  dffv2  5938
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