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Theorem fnressn 5574
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 3508 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
21reseq2d 4789 . . . . 5  |-  ( x  =  B  ->  ( F  |`  { x }
)  =  ( F  |`  { B } ) )
3 fveq2 5389 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4 opeq12 3677 . . . . . . 7  |-  ( ( x  =  B  /\  ( F `  x )  =  ( F `  B ) )  ->  <. x ,  ( F `
 x ) >.  =  <. B ,  ( F `  B )
>. )
53, 4mpdan 417 . . . . . 6  |-  ( x  =  B  ->  <. x ,  ( F `  x ) >.  =  <. B ,  ( F `  B ) >. )
65sneqd 3510 . . . . 5  |-  ( x  =  B  ->  { <. x ,  ( F `  x ) >. }  =  { <. B ,  ( F `  B )
>. } )
72, 6eqeq12d 2132 . . . 4  |-  ( x  =  B  ->  (
( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } 
<->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
87imbi2d 229 . . 3  |-  ( x  =  B  ->  (
( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )  <->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B , 
( F `  B
) >. } ) ) )
9 vex 2663 . . . . . . 7  |-  x  e. 
_V
109snss 3619 . . . . . 6  |-  ( x  e.  A  <->  { x }  C_  A )
11 fnssres 5206 . . . . . 6  |-  ( ( F  Fn  A  /\  { x }  C_  A
)  ->  ( F  |` 
{ x } )  Fn  { x }
)
1210, 11sylan2b 285 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
13 dffn2 5244 . . . . . . 7  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } ) : { x } --> _V )
149fsn2 5562 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> _V  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
1513, 14bitri 183 . . . . . 6  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
16 vsnid 3527 . . . . . . . . . . 11  |-  x  e. 
{ x }
17 fvres 5413 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1816, 17ax-mp 5 . . . . . . . . . 10  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1918opeq2i 3679 . . . . . . . . 9  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
2019sneqi 3509 . . . . . . . 8  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
2120eqeq2i 2128 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
22 snssi 3634 . . . . . . . . . 10  |-  ( x  e.  A  ->  { x }  C_  A )
2322, 11sylan2 284 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
24 funfvex 5406 . . . . . . . . . 10  |-  ( ( Fun  ( F  |`  { x } )  /\  x  e.  dom  ( F  |`  { x } ) )  -> 
( ( F  |`  { x } ) `
 x )  e. 
_V )
2524funfni 5193 . . . . . . . . 9  |-  ( ( ( F  |`  { x } )  Fn  {
x }  /\  x  e.  { x } )  ->  ( ( F  |`  { x } ) `
 x )  e. 
_V )
2623, 16, 25sylancl 409 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) `
 x )  e. 
_V )
2726biantrurd 303 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } 
<->  ( ( ( F  |`  { x } ) `
 x )  e. 
_V  /\  ( F  |` 
{ x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } ) ) )
2821, 27syl5rbbr 194 . . . . . 6  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( ( ( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } )  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } ) )
2915, 28syl5bb 191 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } )  Fn  { x }  <->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
3012, 29mpbid 146 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
3130expcom 115 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
328, 31vtoclga 2726 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
3332impcom 124 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   _Vcvv 2660    C_ wss 3041   {csn 3497   <.cop 3500    |` cres 4511    Fn wfn 5088   -->wf 5089   ` 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-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-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-reu 2400  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  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-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  fressnfv  5575  fnsnsplitss  5587  fnsnsplitdc  6369  dif1en  6741  fnfi  6793  fseq1p1m1  9842  resunimafz0  10542
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