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Theorem fressnfv 5573
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 3506 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
2 reseq2 4782 . . . . . . . 8  |-  ( { x }  =  { B }  ->  ( F  |`  { x } )  =  ( F  |`  { B } ) )
32feq1d 5227 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { x } --> C ) )
4 feq2 5224 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { B } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
53, 4bitrd 187 . . . . . 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 5387 . . . . . 6  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
87eleq1d 2184 . . . . 5  |-  ( x  =  B  ->  (
( F `  x
)  e.  C  <->  ( F `  B )  e.  C
) )
96, 8bibi12d 234 . . . 4  |-  ( x  =  B  ->  (
( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C )  <-> 
( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
109imbi2d 229 . . 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 5572 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
12 vsnid 3525 . . . . . . . . . 10  |-  x  e. 
{ x }
13 fvres 5411 . . . . . . . . . 10  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1412, 13ax-mp 5 . . . . . . . . 9  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1514opeq2i 3677 . . . . . . . 8  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
1615sneqi 3507 . . . . . . 7  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
1716eqeq2i 2126 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
18 vex 2661 . . . . . . . 8  |-  x  e. 
_V
1918fsn2 5560 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> C  <->  ( (
( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
2014eleq1i 2181 . . . . . . . 8  |-  ( ( ( F  |`  { x } ) `  x
)  e.  C  <->  ( F `  x )  e.  C
)
21 iba 296 . . . . . . . 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 194 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( ( ( F  |`  { x } ) `
 x )  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F `  x )  e.  C
) )
2319, 22syl5bb 191 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) )
2417, 23sylbir 134 . . . . 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 115 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) ) )
2710, 26vtoclga 2724 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
2827impcom 124 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   {csn 3495   <.cop 3498    |` cres 4509    Fn wfn 5086   -->wf 5087   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099
This theorem is referenced by: (None)
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