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Theorem nfunsn 5558
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 2183 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4961 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 elsn 3655 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4026 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 187 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 472 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 187 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2190 . . . . . . 7  |-  ( E* y  A F y  ->  E* y  x ( F  |`  { A } ) y )
101, 9syl 15 . . . . . 6  |-  ( E! y  A F y  ->  E* y  x ( F  |`  { A } ) y )
11 tz6.12-2 5516 . . . . . 6  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1210, 11nsyl4 134 . . . . 5  |-  ( -.  ( F `  A
)  =  (/)  ->  E* y  x ( F  |`  { A } ) y )
1312alrimiv 1617 . . . 4  |-  ( -.  ( F `  A
)  =  (/)  ->  A. x E* y  x ( F  |`  { A }
) y )
14 relres 4983 . . . 4  |-  Rel  ( F  |`  { A }
)
1513, 14jctil 523 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
16 dffun6 5270 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1715, 16sylibr 203 . 2  |-  ( -.  ( F `  A
)  =  (/)  ->  Fun  ( F  |`  { A } ) )
1817con1i 121 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144   (/)c0 3455   {csn 3640   class class class wbr 4023    |` cres 4691   Rel wrel 4694   Fun wfun 5249   ` cfv 5255
This theorem is referenced by:  dffv2  5592  fvfundmfvn0  27986
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263
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