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

Theorem fnressn 5381
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 3417 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
21reseq2d 4640 . . . . 5  |-  ( x  =  B  ->  ( F  |`  { x }
)  =  ( F  |`  { B } ) )
3 fveq2 5209 . . . . . . 7  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
4 opeq12 3580 . . . . . . 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 3419 . . . . 5  |-  ( x  =  B  ->  { <. x ,  ( F `  x ) >. }  =  { <. B ,  ( F `  B )
>. } )
72, 6eqeq12d 2096 . . . 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 2605 . . . . . . 7  |-  x  e. 
_V
109snss 3524 . . . . . 6  |-  ( x  e.  A  <->  { x }  C_  A )
11 fnssres 5043 . . . . . 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 5078 . . . . . . 7  |-  ( ( F  |`  { x } )  Fn  {
x }  <->  ( F  |` 
{ x } ) : { x } --> _V )
149fsn2 5369 . . . . . . 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 3434 . . . . . . . . . . 11  |-  x  e. 
{ x }
17 fvres 5230 . . . . . . . . . . 11  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1816, 17ax-mp 7 . . . . . . . . . 10  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1918opeq2i 3582 . . . . . . . . 9  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
2019sneqi 3418 . . . . . . . 8  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
2120eqeq2i 2092 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
22 snssi 3537 . . . . . . . . . 10  |-  ( x  e.  A  ->  { x }  C_  A )
2322, 11sylan2 280 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  Fn  {
x } )
24 funfvex 5223 . . . . . . . . . 10  |-  ( ( Fun  ( F  |`  { x } )  /\  x  e.  dom  ( F  |`  { x } ) )  -> 
( ( F  |`  { x } ) `
 x )  e. 
_V )
2524funfni 5030 . . . . . . . . 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 2665 . 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 1285    e. wcel 1434   _Vcvv 2602    C_ wss 2974   {csn 3406   <.cop 3409    |` cres 4373    Fn wfn 4927   -->wf 4928   ` cfv 4932
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940
This theorem is referenced by:  fressnfv  5382  dif1en  6414  fnfi  6446  fseq1p1m1  9187
  Copyright terms: Public domain W3C validator