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Theorem fnressn 5467
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 3452 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
21reseq2d 4701 . . . . 5  |-  ( x  =  B  ->  ( F  |`  { x }
)  =  ( F  |`  { B } ) )
3 fveq2 5289 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4 opeq12 3619 . . . . . . 7  |-  ( ( x  =  B  /\  ( F `  x )  =  ( F `  B ) )  ->  <. x ,  ( F `
 x ) >.  =  <. B ,  ( F `  B )
>. )
53, 4mpdan 412 . . . . . 6  |-  ( x  =  B  ->  <. x ,  ( F `  x ) >.  =  <. B ,  ( F `  B ) >. )
65sneqd 3454 . . . . 5  |-  ( x  =  B  ->  { <. x ,  ( F `  x ) >. }  =  { <. B ,  ( F `  B )
>. } )
72, 6eqeq12d 2102 . . . 4  |-  ( x  =  B  ->  (
( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } 
<->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
87imbi2d 228 . . 3  |-  ( x  =  B  ->  (
( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )  <->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B , 
( F `  B
) >. } ) ) )
9 vex 2622 . . . . . . 7  |-  x  e. 
_V
109snss 3561 . . . . . 6  |-  ( x  e.  A  <->  { x }  C_  A )
11 fnssres 5113 . . . . . 6  |-  ( ( F  Fn  A  /\  { x }  C_  A
)  ->  ( F  |` 
{ x } )  Fn  { x }
)
1210, 11sylan2b 281 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
13 dffn2 5149 . . . . . . 7  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } ) : { x } --> _V )
149fsn2 5455 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> _V  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
1513, 14bitri 182 . . . . . 6  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( (
( F  |`  { x } ) `  x
)  e.  _V  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
16 vsnid 3471 . . . . . . . . . . 11  |-  x  e. 
{ x }
17 fvres 5313 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1816, 17ax-mp 7 . . . . . . . . . 10  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1918opeq2i 3621 . . . . . . . . 9  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
2019sneqi 3453 . . . . . . . 8  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
2120eqeq2i 2098 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
22 snssi 3576 . . . . . . . . . 10  |-  ( x  e.  A  ->  { x }  C_  A )
2322, 11sylan2 280 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
24 funfvex 5306 . . . . . . . . . 10  |-  ( ( Fun  ( F  |`  { x } )  /\  x  e.  dom  ( F  |`  { x } ) )  -> 
( ( F  |`  { x } ) `
 x )  e. 
_V )
2524funfni 5100 . . . . . . . . 9  |-  ( ( ( F  |`  { x } )  Fn  {
x }  /\  x  e.  { x } )  ->  ( ( F  |`  { x } ) `
 x )  e. 
_V )
2623, 16, 25sylancl 404 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) `
 x )  e. 
_V )
2726biantrurd 299 . . . . . . 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 193 . . . . . 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 190 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } )  Fn  { x }  <->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
3012, 29mpbid 145 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
3130expcom 114 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } ) )
328, 31vtoclga 2685 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } ) )
3332impcom 123 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    C_ wss 2997   {csn 3441   <.cop 3444    |` cres 4430    Fn wfn 4997   -->wf 4998   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010
This theorem is referenced by:  fressnfv  5468  dif1en  6575  fnfi  6625  fseq1p1m1  9475  resunimafz0  10201
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