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Theorem fnressn 5889
Description: A function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fnressn  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )

Proof of Theorem fnressn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3882 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
21reseq2d 5105 . . . . 5  |-  ( x  =  B  ->  ( F  |`  { x }
)  =  ( F  |`  { B } ) )
3 fveq2 5686 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4 opeq12 4056 . . . . . . 7  |-  ( ( x  =  B  /\  ( F `  x )  =  ( F `  B ) )  ->  <. x ,  ( F `
 x ) >.  =  <. B ,  ( F `  B )
>. )
53, 4mpdan 668 . . . . . 6  |-  ( x  =  B  ->  <. x ,  ( F `  x ) >.  =  <. B ,  ( F `  B ) >. )
65sneqd 3884 . . . . 5  |-  ( x  =  B  ->  { <. x ,  ( F `  x ) >. }  =  { <. B ,  ( F `  B )
>. } )
72, 6eqeq12d 2452 . . . 4  |-  ( x  =  B  ->  (
( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } 
<->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
87imbi2d 316 . . 3  |-  ( x  =  B  ->  (
( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )  <->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B , 
( F `  B
) >. } ) ) )
9 vex 2970 . . . . . . 7  |-  x  e. 
_V
109snss 3994 . . . . . 6  |-  ( x  e.  A  <->  { x }  C_  A )
11 fnssres 5519 . . . . . 6  |-  ( ( F  Fn  A  /\  { x }  C_  A
)  ->  ( F  |` 
{ x } )  Fn  { x }
)
1210, 11sylan2b 475 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
13 dffn2 5555 . . . . . 6  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } ) : { x } --> _V )
149fsn2 5878 . . . . . 6  |-  ( ( F  |`  { x } ) : {
x } --> _V  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
15 fvex 5696 . . . . . . . 8  |-  ( ( F  |`  { x } ) `  x
)  e.  _V
1615biantrur 506 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
17 ssnid 3901 . . . . . . . . . . 11  |-  x  e. 
{ x }
18 fvres 5699 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1917, 18ax-mp 5 . . . . . . . . . 10  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
2019opeq2i 4058 . . . . . . . . 9  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
2120sneqi 3883 . . . . . . . 8  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
2221eqeq2i 2448 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2316, 22bitr3i 251 . . . . . 6  |-  ( ( ( ( F  |`  { x } ) `
 x )  e. 
_V  /\  ( F  |` 
{ x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2413, 14, 233bitri 271 . . . . 5  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2512, 24sylib 196 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
2625expcom 435 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
278, 26vtoclga 3031 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
2827impcom 430 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    C_ wss 3323   {csn 3872   <.cop 3878    |` cres 4837    Fn wfn 5408   -->wf 5409   ` cfv 5413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
This theorem is referenced by:  funressn  5890  fressnfv  5891  fnsnsplit  5910  canthp1lem2  8812  fseq1p1m1  11526  dprd2da  16529  dmdprdpr  16536  dprdpr  16537  dpjlem  16538  pgpfaclem1  16570  islindf4  18242  xpstopnlem1  19357  ptcmpfi  19361  2pthlem1  23445  eupath2lem3  23551  ginvsn  23787  subfacp1lem5  27024  cvmliftlem10  27135
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