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Theorem fressnfv 5350
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 3386 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
2 reseq2 4607 . . . . . . . 8  |-  ( { x }  =  { B }  ->  ( F  |`  { x } )  =  ( F  |`  { B } ) )
32feq1d 5034 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { x } --> C ) )
4 feq2 5031 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { B } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
53, 4bitrd 177 . . . . . 6  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
61, 5syl 14 . . . . 5  |-  ( x  =  B  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
7 fveq2 5178 . . . . . 6  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
87eleq1d 2106 . . . . 5  |-  ( x  =  B  ->  (
( F `  x
)  e.  C  <->  ( F `  B )  e.  C
) )
96, 8bibi12d 224 . . . 4  |-  ( x  =  B  ->  (
( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C )  <-> 
( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
109imbi2d 219 . . 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 5349 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
12 vsnid 3403 . . . . . . . . . 10  |-  x  e. 
{ x }
13 fvres 5198 . . . . . . . . . 10  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1412, 13ax-mp 7 . . . . . . . . 9  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1514opeq2i 3553 . . . . . . . 8  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
1615sneqi 3387 . . . . . . 7  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
1716eqeq2i 2050 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
18 vex 2560 . . . . . . . 8  |-  x  e. 
_V
1918fsn2 5337 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> C  <->  ( (
( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
2014eleq1i 2103 . . . . . . . 8  |-  ( ( ( F  |`  { x } ) `  x
)  e.  C  <->  ( F `  x )  e.  C
)
21 iba 284 . . . . . . . 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 184 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( ( ( F  |`  { x } ) `
 x )  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F `  x )  e.  C
) )
2319, 22syl5bb 181 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) )
2417, 23sylbir 125 . . . . 5  |-  ( ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. }  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2511, 24syl 14 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2625expcom 109 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) ) )
2710, 26vtoclga 2619 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
2827impcom 116 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    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {csn 3375   <.cop 3378    |` cres 4347    Fn wfn 4897   -->wf 4898   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910
This theorem is referenced by: (None)
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