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Theorem fnressn 5721
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 3664 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
21reseq2d 4971 . . . . 5  |-  ( x  =  B  ->  ( F  |`  { x }
)  =  ( F  |`  { B } ) )
3 fveq2 5541 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4 opeq12 3814 . . . . . . 7  |-  ( ( x  =  B  /\  ( F `  x )  =  ( F `  B ) )  ->  <. x ,  ( F `
 x ) >.  =  <. B ,  ( F `  B )
>. )
53, 4mpdan 649 . . . . . 6  |-  ( x  =  B  ->  <. x ,  ( F `  x ) >.  =  <. B ,  ( F `  B ) >. )
65sneqd 3666 . . . . 5  |-  ( x  =  B  ->  { <. x ,  ( F `  x ) >. }  =  { <. B ,  ( F `  B )
>. } )
72, 6eqeq12d 2310 . . . 4  |-  ( x  =  B  ->  (
( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } 
<->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
87imbi2d 307 . . 3  |-  ( x  =  B  ->  (
( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )  <->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B , 
( F `  B
) >. } ) ) )
9 vex 2804 . . . . . . 7  |-  x  e. 
_V
109snss 3761 . . . . . 6  |-  ( x  e.  A  <->  { x }  C_  A )
11 fnssres 5373 . . . . . 6  |-  ( ( F  Fn  A  /\  { x }  C_  A
)  ->  ( F  |` 
{ x } )  Fn  { x }
)
1210, 11sylan2b 461 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
13 dffn2 5406 . . . . . 6  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } ) : { x } --> _V )
149fsn2 5714 . . . . . 6  |-  ( ( F  |`  { x } ) : {
x } --> _V  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
15 fvex 5555 . . . . . . . 8  |-  ( ( F  |`  { x } ) `  x
)  e.  _V
1615biantrur 492 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
179snid 3680 . . . . . . . . . . 11  |-  x  e. 
{ x }
18 fvres 5558 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1917, 18ax-mp 8 . . . . . . . . . 10  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
2019opeq2i 3816 . . . . . . . . 9  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
2120sneqi 3665 . . . . . . . 8  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
2221eqeq2i 2306 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2316, 22bitr3i 242 . . . . . 6  |-  ( ( ( ( F  |`  { x } ) `
 x )  e. 
_V  /\  ( F  |` 
{ x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2413, 14, 233bitri 262 . . . . 5  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
2512, 24sylib 188 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
2625expcom 424 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
278, 26vtoclga 2862 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
2827impcom 419 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    C_ wss 3165   {csn 3653   <.cop 3656    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271
This theorem is referenced by:  funressn  5722  fressnfv  5723  fnsnsplit  5733  canthp1lem2  8291  fseq1p1m1  10873  dprd2da  15293  dmdprdpr  15300  dprdpr  15301  dpjlem  15302  pgpfaclem1  15332  xpstopnlem1  17516  ptcmpfi  17520  ginvsn  21032  subfacp1lem5  23730  cvmliftlem10  23840  eupath2lem3  23918  islindf4  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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