ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfunsn Unicode version

Theorem nfunsn 5423
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 2009 . . . . . . 7  |-  ( E! y  A F y  ->  E* y  A F y )
2 vex 2663 . . . . . . . . . 10  |-  y  e. 
_V
32brres 4795 . . . . . . . . 9  |-  ( x ( F  |`  { A } ) y  <->  ( x F y  /\  x  e.  { A } ) )
4 velsn 3514 . . . . . . . . . . 11  |-  ( x  e.  { A }  <->  x  =  A )
5 breq1 3902 . . . . . . . . . . 11  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
64, 5sylbi 120 . . . . . . . . . 10  |-  ( x  e.  { A }  ->  ( x F y  <-> 
A F y ) )
76biimpac 296 . . . . . . . . 9  |-  ( ( x F y  /\  x  e.  { A } )  ->  A F y )
83, 7sylbi 120 . . . . . . . 8  |-  ( x ( F  |`  { A } ) y  ->  A F y )
98moimi 2042 . . . . . . 7  |-  ( E* y  A F y  ->  E* y  x ( F  |`  { A } ) y )
101, 9syl 14 . . . . . 6  |-  ( E! y  A F y  ->  E* y  x ( F  |`  { A } ) y )
1110alrimiv 1830 . . . . 5  |-  ( E! y  A F y  ->  A. x E* y  x ( F  |`  { A } ) y )
12 relres 4817 . . . . 5  |-  Rel  ( F  |`  { A }
)
1311, 12jctil 310 . . . 4  |-  ( E! y  A F y  ->  ( Rel  ( F  |`  { A }
)  /\  A. x E* y  x ( F  |`  { A }
) y ) )
14 dffun6 5107 . . . 4  |-  ( Fun  ( F  |`  { A } )  <->  ( Rel  ( F  |`  { A } )  /\  A. x E* y  x ( F  |`  { A } ) y ) )
1513, 14sylibr 133 . . 3  |-  ( E! y  A F y  ->  Fun  ( F  |` 
{ A } ) )
1615con3i 606 . 2  |-  ( -. 
Fun  ( F  |`  { A } )  ->  -.  E! y  A F y )
17 tz6.12-2 5380 . 2  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
1816, 17syl 14 1  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   E!weu 1977   E*wmo 1978   (/)c0 3333   {csn 3497   class class class wbr 3899    |` cres 4511   Rel wrel 4514   Fun wfun 5087   ` cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-res 4521  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator