MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funressn Structured version   Unicode version

Theorem funressn 6085
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 5623 . . . 4  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 fnressn 6084 . . . 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 3551 . . 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 4092 . . . . 5  |-  ( ( dom  F  i^i  { B } )  =  (/)  <->  -.  B  e.  dom  F )
7 fnresdisj 5697 . . . . . 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 3823 . . 3  |-  (/)  C_  { <. B ,  ( F `  B ) >. }
1210, 11syl6eqss 3549 . 2  |-  ( ( Fun  F  /\  -.  B  e.  dom  F )  ->  ( F  |`  { B } )  C_  {
<. B ,  ( F `
 B ) >. } )
135, 12pm2.61dan 791 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 1395    e. wcel 1819    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   <.cop 4038   dom cdm 5008    |` cres 5010   Fun wfun 5588    Fn wfn 5589   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  fnsnb  6091  tfrlem16  7080  fnfi  7816  fodomfi  7817  bnj142OLD  33882
  Copyright terms: Public domain W3C validator