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Theorem funressn 5706
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 5283 . . . 4  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnressn 5705 . . . 4  |-  ( ( F  Fn  dom  F  /\  B  e.  dom  F )  ->  ( F  |` 
{ B } )  =  { <. B , 
( F `  B
) >. } )
31, 2sylanb 458 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
4 eqimss 3230 . . 3  |-  ( ( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. }  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
53, 4syl 15 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F  |`  { B } )  C_  { <. B ,  ( F `  B ) >. } )
6 disjsn 3693 . . . . 5  |-  ( ( dom  F  i^i  { B } )  =  (/)  <->  -.  B  e.  dom  F )
7 fnresdisj 5354 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B } )  =  (/) ) )
81, 7sylbi 187 . . . . 5  |-  ( Fun 
F  ->  ( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B }
)  =  (/) ) )
96, 8syl5bbr 250 . . . 4  |-  ( Fun 
F  ->  ( -.  B  e.  dom  F  <->  ( F  |` 
{ B } )  =  (/) ) )
109biimpa 470 . . 3  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  =  (/) )
11 0ss 3483 . . . 4  |-  (/)  C_  { <. B ,  ( F `  B ) >. }
1211a1i 10 . . 3  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  (/)  C_  { <. B , 
( F `  B
) >. } )
1310, 12eqsstrd 3212 . 2  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
145, 13pm2.61dan 766 1  |-  ( Fun 
F  ->  ( F  |` 
{ B } ) 
C_  { <. B , 
( F `  B
) >. } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   <.cop 3643   dom cdm 4689    |` cres 4691   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tfrlem16  6409  fnfi  7134  fodomfi  7135  bnj142  28754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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