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Theorem funcoressn 28095
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 4992 . . . . . . . . 9  |-  dom  ( F  |`  { ( G `
 X ) } )  =  ( { ( G `  X
) }  i^i  dom  F )
2 snssi 3775 . . . . . . . . . 10  |-  ( ( G `  X )  e.  dom  F  ->  { ( G `  X ) }  C_  dom  F )
3 df-ss 3179 . . . . . . . . . 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 2340 . . . . . . . 8  |-  ( ( G `  X )  e.  dom  F  ->  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } )
6 df-fn 5274 . . . . . . . . 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 5598 . . . . . . . . 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 4718 . . . . . . . 8  |-  ( G
" { X }
)  =  ran  ( G  |`  { X }
)
1412, 13syl6eq 2344 . . . . . . 7  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ran  ( G  |`  { X } ) )
1514reseq2d 4971 . . . . . 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 5353 . . . . 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 5357 . . . . . . 7  |-  ( G  Fn  A  ->  Fun  G )
19 funres 5309 . . . . . . . 8  |-  ( Fun 
G  ->  Fun  ( G  |`  { X } ) )
20 funfn 5299 . . . . . . . 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 28094 . . . 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 5357 . . 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 5193 . . 3  |-  ( ( F  o.  G )  |`  { X } )  =  ( F  o.  ( G  |`  { X } ) )
3029funeqi 5291 . 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   {csn 3653   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  afvco2  28144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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