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Theorem fressnfv 6073
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 4037 . . . . . 6  |-  ( x  =  B  ->  { x }  =  { B } )
2 reseq2 5266 . . . . . . . 8  |-  ( { x }  =  { B }  ->  ( F  |`  { x } )  =  ( F  |`  { B } ) )
32feq1d 5715 . . . . . . 7  |-  ( { x }  =  { B }  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { x } --> C ) )
4 feq2 5712 . . . . . . 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 16 . . . . 5  |-  ( x  =  B  ->  (
( F  |`  { x } ) : {
x } --> C  <->  ( F  |` 
{ B } ) : { B } --> C ) )
7 fveq2 5864 . . . . . 6  |-  ( x  =  B  ->  ( F `  x )  =  ( F `  B ) )
87eleq1d 2536 . . . . 5  |-  ( x  =  B  ->  (
( F `  x
)  e.  C  <->  ( F `  B )  e.  C
) )
96, 8bibi12d 321 . . . 4  |-  ( x  =  B  ->  (
( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C )  <-> 
( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
109imbi2d 316 . . 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 6071 . . . . 5  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( F  |`  { x } )  =  { <. x ,  ( F `
 x ) >. } )
12 ssnid 4056 . . . . . . . . . 10  |-  x  e. 
{ x }
13 fvres 5878 . . . . . . . . . 10  |-  ( x  e.  { x }  ->  ( ( F  |`  { x } ) `
 x )  =  ( F `  x
) )
1412, 13ax-mp 5 . . . . . . . . 9  |-  ( ( F  |`  { x } ) `  x
)  =  ( F `
 x )
1514opeq2i 4217 . . . . . . . 8  |-  <. x ,  ( ( F  |`  { x } ) `
 x ) >.  =  <. x ,  ( F `  x )
>.
1615sneqi 4038 . . . . . . 7  |-  { <. x ,  ( ( F  |`  { x } ) `
 x ) >. }  =  { <. x ,  ( F `  x ) >. }
1716eqeq2i 2485 . . . . . 6  |-  ( ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. }  <->  ( F  |` 
{ x } )  =  { <. x ,  ( F `  x ) >. } )
18 vex 3116 . . . . . . . 8  |-  x  e. 
_V
1918fsn2 6059 . . . . . . 7  |-  ( ( F  |`  { x } ) : {
x } --> C  <->  ( (
( F  |`  { x } ) `  x
)  e.  C  /\  ( F  |`  { x } )  =  { <. x ,  ( ( F  |`  { x } ) `  x
) >. } ) )
2014eleq1i 2544 . . . . . . . 8  |-  ( ( ( F  |`  { x } ) `  x
)  e.  C  <->  ( F `  x )  e.  C
)
21 iba 503 . . . . . . . 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 16 . . . 4  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F  |`  { x } ) : { x } --> C 
<->  ( F `  x
)  e.  C ) )
2625expcom 435 . . 3  |-  ( x  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { x } ) : {
x } --> C  <->  ( F `  x )  e.  C
) ) )
2710, 26vtoclga 3177 . 2  |-  ( B  e.  A  ->  ( F  Fn  A  ->  ( ( F  |`  { B } ) : { B } --> C  <->  ( F `  B )  e.  C
) ) )
2827impcom 430 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 369    = wceq 1379    e. wcel 1767   {csn 4027   <.cop 4033    |` cres 5001    Fn wfn 5581   -->wf 5582   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594
This theorem is referenced by: (None)
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