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Theorem fressnfv 6021
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 3981 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
2 reseq2 5210 . . . . . . . 8  |-  ( { x }  =  { B }  ->  ( F  |`  { x } )  =  ( F  |`  { B } ) )
32feq1d 5656 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { x } --> C ) )
4 feq2 5653 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { B } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
53, 4bitrd 253 . . . . . 6  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
61, 5syl 17 . . . . 5  |-  ( x  =  B  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
7 fveq2 5805 . . . . . 6  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
87eleq1d 2471 . . . . 5  |-  ( x  =  B  ->  (
( F `  x
)  e.  C  <->  ( F `  B )  e.  C
) )
96, 8bibi12d 319 . . . 4  |-  ( x  =  B  ->  (
( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C )  <-> 
( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
109imbi2d 314 . . 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 6019 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
12 ssnid 4000 . . . . . . . . . 10  |-  x  e. 
{ x }
13 fvres 5819 . . . . . . . . . 10  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1412, 13ax-mp 5 . . . . . . . . 9  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1514opeq2i 4162 . . . . . . . 8  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
1615sneqi 3982 . . . . . . 7  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
1716eqeq2i 2420 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
18 vex 3061 . . . . . . . 8  |-  x  e. 
_V
1918fsn2 6005 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> C  <->  ( (
( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
2014eleq1i 2479 . . . . . . . 8  |-  ( ( ( F  |`  { x } ) `  x
)  e.  C  <->  ( F `  x )  e.  C
)
21 iba 501 . . . . . . . 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 260 . . . . . . 7  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( ( ( F  |`  { x } ) `
 x )  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. } )  <->  ( F `  x )  e.  C
) )
2319, 22syl5bb 257 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) )
2417, 23sylbir 213 . . . . 5  |-  ( ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. }  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2511, 24syl 17 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2625expcom 433 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) ) )
2710, 26vtoclga 3122 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
2827impcom 428 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   {csn 3971   <.cop 3977    |` cres 4944    Fn wfn 5520   -->wf 5521   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533
This theorem is referenced by: (None)
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