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Theorem nfunsn 5823
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 2293 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 3074 . . . . . . . . . 10  |-  y  e. 
_V
32brres 5218 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 elsn 3992 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4396 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 195 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 486 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 195 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2327 . . . . . . 7  |-  ( E* y  A F y  ->  E* y  x ( F  |`  { A } ) y )
101, 9syl 16 . . . . . 6  |-  ( E! y  A F y  ->  E* y  x ( F  |`  { A } ) y )
11 tz6.12-2 5783 . . . . . 6  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1210, 11nsyl4 142 . . . . 5  |-  ( -.  ( F `  A
)  =  (/)  ->  E* y  x ( F  |`  { A } ) y )
1312alrimiv 1686 . . . 4  |-  ( -.  ( F `  A
)  =  (/)  ->  A. x E* y  x ( F  |`  { A }
) y )
14 relres 5239 . . . 4  |-  Rel  ( F  |`  { A }
)
1513, 14jctil 537 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
16 dffun6 5534 . . 3  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1715, 16sylibr 212 . 2  |-  ( -.  ( F `  A
)  =  (/)  ->  Fun  ( F  |`  { A } ) )
1817con1i 129 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368    = wceq 1370    e. wcel 1758   E!weu 2260   E*wmo 2261   (/)c0 3738   {csn 3978   class class class wbr 4393    |` cres 4943   Rel wrel 4946   Fun wfun 5513   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-res 4953  df-iota 5482  df-fun 5521  df-fv 5527
This theorem is referenced by:  fvfundmfvn0  5824  dffv2  5866
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