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Theorem funressn 6072
Description: A function restricted to a singleton. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
funressn  |-  ( Fun 
F  ->  ( F  |` 
{ B } ) 
C_  { <. B , 
( F `  B
) >. } )

Proof of Theorem funressn
StepHypRef Expression
1 funfn 5615 . . . 4  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnressn 6071 . . . 4  |-  ( ( F  Fn  dom  F  /\  B  e.  dom  F )  ->  ( F  |` 
{ B } )  =  { <. B , 
( F `  B
) >. } )
31, 2sylanb 472 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
4 eqimss 3556 . . 3  |-  ( ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. }  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
53, 4syl 16 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F  |`  { B } )  C_  { <. B ,  ( F `  B ) >. } )
6 disjsn 4088 . . . . 5  |-  ( ( dom  F  i^i  { B } )  =  (/)  <->  -.  B  e.  dom  F )
7 fnresdisj 5689 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B } )  =  (/) ) )
81, 7sylbi 195 . . . . 5  |-  ( Fun 
F  ->  ( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B }
)  =  (/) ) )
96, 8syl5bbr 259 . . . 4  |-  ( Fun 
F  ->  ( -.  B  e.  dom  F  <->  ( F  |` 
{ B } )  =  (/) ) )
109biimpa 484 . . 3  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  =  (/) )
11 0ss 3814 . . 3  |-  (/)  C_  { <. B ,  ( F `  B ) >. }
1210, 11syl6eqss 3554 . 2  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
135, 12pm2.61dan 789 1  |-  ( Fun 
F  ->  ( F  |` 
{ B } ) 
C_  { <. B , 
( F `  B
) >. } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   dom cdm 4999    |` cres 5001   Fun wfun 5580    Fn wfn 5581   ` 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:  fnsnb  6078  tfrlem16  7059  fnfi  7794  fodomfi  7795  bnj142OLD  32861
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