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Theorem fressnfv 5707
Description: The value of a function restricted to a singleton. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
fressnfv  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) )

Proof of Theorem fressnfv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3651 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
2 reseq2 4950 . . . . . . . 8  |-  ( { x }  =  { B }  ->  ( F  |`  { x } )  =  ( F  |`  { B } ) )
32feq1d 5379 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { x } --> C ) )
4 feq2 5376 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { B } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
53, 4bitrd 244 . . . . . 6  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
61, 5syl 15 . . . . 5  |-  ( x  =  B  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
7 fveq2 5525 . . . . . 6  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
87eleq1d 2349 . . . . 5  |-  ( x  =  B  ->  (
( F `  x
)  e.  C  <->  ( F `  B )  e.  C
) )
96, 8bibi12d 312 . . . 4  |-  ( x  =  B  ->  (
( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C )  <-> 
( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
109imbi2d 307 . . 3  |-  ( x  =  B  ->  (
( F  Fn  A  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )  <->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C 
<->  ( F `  B
)  e.  C ) ) ) )
11 fnressn 5705 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
12 vex 2791 . . . . . . . . . . 11  |-  x  e. 
_V
1312snid 3667 . . . . . . . . . 10  |-  x  e. 
{ x }
14 fvres 5542 . . . . . . . . . 10  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1513, 14ax-mp 8 . . . . . . . . 9  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1615opeq2i 3800 . . . . . . . 8  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
1716sneqi 3652 . . . . . . 7  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
1817eqeq2i 2293 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
1912fsn2 5698 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> C  <->  ( (
( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
2015eleq1i 2346 . . . . . . . 8  |-  ( ( ( F  |`  { x } ) `  x
)  e.  C  <->  ( F `  x )  e.  C
)
21 iba 489 . . . . . . . 8  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( ( F  |`  { x } ) `
 x )  e.  C  <->  ( ( ( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) ) )
2220, 21syl5rbbr 251 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( ( ( F  |`  { x } ) `
 x )  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F `  x )  e.  C
) )
2319, 22syl5bb 248 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) )
2418, 23sylbir 204 . . . . 5  |-  ( ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. }  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2511, 24syl 15 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2625expcom 424 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) ) )
2710, 26vtoclga 2849 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
2827impcom 419 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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