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Theorem funcoressn 27990
Description: A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
funcoressn  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )

Proof of Theorem funcoressn
StepHypRef Expression
1 dmres 4976 . . . . . . . . 9  |-  dom  ( F  |`  { ( G `
 X ) } )  =  ( { ( G `  X
) }  i^i  dom  F )
2 snssi 3759 . . . . . . . . . 10  |-  ( ( G `  X )  e.  dom  F  ->  { ( G `  X ) }  C_  dom  F )
3 df-ss 3166 . . . . . . . . . 10  |-  ( { ( G `  X
) }  C_  dom  F  <-> 
( { ( G `
 X ) }  i^i  dom  F )  =  { ( G `  X ) } )
42, 3sylib 188 . . . . . . . . 9  |-  ( ( G `  X )  e.  dom  F  -> 
( { ( G `
 X ) }  i^i  dom  F )  =  { ( G `  X ) } )
51, 4syl5eq 2327 . . . . . . . 8  |-  ( ( G `  X )  e.  dom  F  ->  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } )
6 df-fn 5258 . . . . . . . . 9  |-  ( ( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) }  <->  ( Fun  ( F  |`  { ( G `  X ) } )  /\  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } ) )
76simplbi2com 1364 . . . . . . . 8  |-  ( dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) }  ->  ( Fun  ( F  |`  { ( G `  X ) } )  ->  ( F  |`  { ( G `
 X ) } )  Fn  { ( G `  X ) } ) )
85, 7syl 15 . . . . . . 7  |-  ( ( G `  X )  e.  dom  F  -> 
( Fun  ( F  |` 
{ ( G `  X ) } )  ->  ( F  |`  { ( G `  X ) } )  Fn  { ( G `
 X ) } ) )
98imp 418 . . . . . 6  |-  ( ( ( G `  X
)  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  ->  ( F  |` 
{ ( G `  X ) } )  Fn  { ( G `
 X ) } )
109adantr 451 . . . . 5  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) } )
11 fnsnfv 5582 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
1211adantl 452 . . . . . . . 8  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ( G " { X } ) )
13 df-ima 4702 . . . . . . . 8  |-  ( G
" { X }
)  =  ran  ( G  |`  { X }
)
1412, 13syl6eq 2331 . . . . . . 7  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ran  ( G  |`  { X } ) )
1514reseq2d 4955 . . . . . 6  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  { ( G `  X ) } )  =  ( F  |`  ran  ( G  |`  { X } ) ) )
1615, 14fneq12d 5337 . . . . 5  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( ( F  |`  { ( G `  X ) } )  Fn  { ( G `
 X ) }  <-> 
( F  |`  ran  ( G  |`  { X }
) )  Fn  ran  ( G  |`  { X } ) ) )
1710, 16mpbid 201 . . . 4  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  ran  ( G  |`  { X }
) )  Fn  ran  ( G  |`  { X } ) )
18 fnfun 5341 . . . . . . 7  |-  ( G  Fn  A  ->  Fun  G )
19 funres 5293 . . . . . . . 8  |-  ( Fun 
G  ->  Fun  ( G  |`  { X } ) )
20 funfn 5283 . . . . . . . 8  |-  ( Fun  ( G  |`  { X } )  <->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
2119, 20sylib 188 . . . . . . 7  |-  ( Fun 
G  ->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
2218, 21syl 15 . . . . . 6  |-  ( G  Fn  A  ->  ( G  |`  { X }
)  Fn  dom  ( G  |`  { X }
) )
2322adantr 451 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
2423adantl 452 . . . 4  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
25 fnresfnco 27989 . . . 4  |-  ( ( ( F  |`  ran  ( G  |`  { X }
) )  Fn  ran  ( G  |`  { X } )  /\  ( G  |`  { X }
)  Fn  dom  ( G  |`  { X }
) )  ->  ( F  o.  ( G  |` 
{ X } ) )  Fn  dom  ( G  |`  { X }
) )
2617, 24, 25syl2anc 642 . . 3  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  o.  ( G  |`  { X }
) )  Fn  dom  ( G  |`  { X } ) )
27 fnfun 5341 . . 3  |-  ( ( F  o.  ( G  |`  { X } ) )  Fn  dom  ( G  |`  { X }
)  ->  Fun  ( F  o.  ( G  |`  { X } ) ) )
2826, 27syl 15 . 2  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  o.  ( G  |`  { X }
) ) )
29 resco 5177 . . 3  |-  ( ( F  o.  G )  |`  { X } )  =  ( F  o.  ( G  |`  { X } ) )
3029funeqi 5275 . 2  |-  ( Fun  ( ( F  o.  G )  |`  { X } )  <->  Fun  ( F  o.  ( G  |`  { X } ) ) )
3128, 30sylibr 203 1  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   {csn 3640   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    o. ccom 4693   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  afvco2  28037
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-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-fv 5263
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