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Theorem funressn 5882
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 5445 . . . 4  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnressn 5881 . . . 4  |-  ( ( F  Fn  dom  F  /\  B  e.  dom  F )  ->  ( F  |` 
{ B } )  =  { <. B , 
( F `  B
) >. } )
31, 2sylanb 459 . . 3  |-  ( ( Fun  F  /\  B  e.  dom  F )  -> 
( F  |`  { B } )  =  { <. B ,  ( F `
 B ) >. } )
4 eqimss 3364 . . 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 3832 . . . . 5  |-  ( ( dom  F  i^i  { B } )  =  (/)  <->  -.  B  e.  dom  F )
7 fnresdisj 5518 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B } )  =  (/) ) )
81, 7sylbi 188 . . . . 5  |-  ( Fun 
F  ->  ( ( dom  F  i^i  { B } )  =  (/)  <->  ( F  |`  { B }
)  =  (/) ) )
96, 8syl5bbr 251 . . . 4  |-  ( Fun 
F  ->  ( -.  B  e.  dom  F  <->  ( F  |` 
{ B } )  =  (/) ) )
109biimpa 471 . . 3  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  =  (/) )
11 0ss 3620 . . 3  |-  (/)  C_  { <. B ,  ( F `  B ) >. }
1210, 11syl6eqss 3362 . 2  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
135, 12pm2.61dan 767 1  |-  ( Fun 
F  ->  ( F  |` 
{ B } ) 
C_  { <. B , 
( F `  B
) >. } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   (/)c0 3592   {csn 3778   <.cop 3781   dom cdm 4841    |` cres 4843   Fun wfun 5411    Fn wfn 5412   ` cfv 5417
This theorem is referenced by:  tfrlem16  6617  fnfi  7347  fodomfi  7348  bnj142  28803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
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