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Theorem nfunsn 5910
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 2348 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 3034 . . . . . . . . . 10  |-  y  e. 
_V
32brres 5117 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 elsn 3973 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4398 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 200 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 494 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 200 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2369 . . . . . . 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 5870 . . . . . 6  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1210, 11nsyl4 149 . . . . 5  |-  ( -.  ( F `  A
)  =  (/)  ->  E* y  x ( F  |`  { A } ) y )
1312alrimiv 1781 . . . 4  |-  ( -.  ( F `  A
)  =  (/)  ->  A. x E* y  x ( F  |`  { A }
) y )
14 relres 5138 . . . 4  |-  Rel  ( F  |`  { A }
)
1513, 14jctil 546 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
16 dffun6 5604 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1715, 16sylibr 217 . 2  |-  ( -.  ( F `  A
)  =  (/)  ->  Fun  ( F  |`  { A } ) )
1817con1i 134 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   E!weu 2319   E*wmo 2320   (/)c0 3722   {csn 3959   class class class wbr 4395    |` cres 4841   Rel wrel 4844   Fun wfun 5583   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597
This theorem is referenced by:  fvfundmfvn0  5911  dffv2  5953
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