MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfunsn Unicode version

Theorem nfunsn 5641
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 2249 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2867 . . . . . . . . . 10  |-  y  e. 
_V
32brres 5043 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 elsn 3731 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 4107 . . . . . . . . . . 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 2256 . . . . . . 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 5599 . . . . . 6  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1210, 11nsyl4 134 . . . . 5  |-  ( -.  ( F `  A
)  =  (/)  ->  E* y  x ( F  |`  { A } ) y )
1312alrimiv 1631 . . . 4  |-  ( -.  ( F `  A
)  =  (/)  ->  A. x E* y  x ( F  |`  { A }
) y )
14 relres 5065 . . . 4  |-  Rel  ( F  |`  { A }
)
1513, 14jctil 523 . . 3  |-  ( -.  ( F `  A
)  =  (/)  ->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
16 dffun6 5352 . . 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 1540    = wceq 1642    e. wcel 1710   E!weu 2209   E*wmo 2210   (/)c0 3531   {csn 3716   class class class wbr 4104    |` cres 4773   Rel wrel 4776   Fun wfun 5331   ` cfv 5337
This theorem is referenced by:  dffv2  5675  fvfundmfvn0  27311
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-res 4783  df-iota 5301  df-fun 5339  df-fv 5345
  Copyright terms: Public domain W3C validator