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Theorem funcoressn 27662
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 dmressnsn 27655 . . . . . . . 8  |-  ( ( G `  X )  e.  dom  F  ->  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } )
2 df-fn 5399 . . . . . . . . 9  |-  ( ( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) }  <->  ( Fun  ( F  |`  { ( G `  X ) } )  /\  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } ) )
32simplbi2com 1380 . . . . . . . 8  |-  ( dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) }  ->  ( Fun  ( F  |`  { ( G `  X ) } )  ->  ( F  |`  { ( G `
 X ) } )  Fn  { ( G `  X ) } ) )
41, 3syl 16 . . . . . . 7  |-  ( ( G `  X )  e.  dom  F  -> 
( Fun  ( F  |` 
{ ( G `  X ) } )  ->  ( F  |`  { ( G `  X ) } )  Fn  { ( G `
 X ) } ) )
54imp 419 . . . . . 6  |-  ( ( ( G `  X
)  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  ->  ( F  |` 
{ ( G `  X ) } )  Fn  { ( G `
 X ) } )
65adantr 452 . . . . 5  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) } )
7 fnsnfv 5727 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
87adantl 453 . . . . . . . 8  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ( G " { X } ) )
9 df-ima 4833 . . . . . . . 8  |-  ( G
" { X }
)  =  ran  ( G  |`  { X }
)
108, 9syl6eq 2437 . . . . . . 7  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ran  ( G  |`  { X } ) )
1110reseq2d 5088 . . . . . 6  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  { ( G `  X ) } )  =  ( F  |`  ran  ( G  |`  { X } ) ) )
1211, 10fneq12d 5480 . . . . 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 } ) ) )
136, 12mpbid 202 . . . 4  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  ran  ( G  |`  { X }
) )  Fn  ran  ( G  |`  { X } ) )
14 fnfun 5484 . . . . . . 7  |-  ( G  Fn  A  ->  Fun  G )
15 funres 5434 . . . . . . . 8  |-  ( Fun 
G  ->  Fun  ( G  |`  { X } ) )
16 funfn 5424 . . . . . . . 8  |-  ( Fun  ( G  |`  { X } )  <->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
1715, 16sylib 189 . . . . . . 7  |-  ( Fun 
G  ->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
1814, 17syl 16 . . . . . 6  |-  ( G  Fn  A  ->  ( G  |`  { X }
)  Fn  dom  ( G  |`  { X }
) )
1918adantr 452 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
2019adantl 453 . . . 4  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
21 fnresfnco 27661 . . . 4  |-  ( ( ( F  |`  ran  ( G  |`  { X }
) )  Fn  ran  ( G  |`  { X } )  /\  ( G  |`  { X }
)  Fn  dom  ( G  |`  { X }
) )  ->  ( F  o.  ( G  |` 
{ X } ) )  Fn  dom  ( G  |`  { X }
) )
2213, 20, 21syl2anc 643 . . 3  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  o.  ( G  |`  { X }
) )  Fn  dom  ( G  |`  { X } ) )
23 fnfun 5484 . . 3  |-  ( ( F  o.  ( G  |`  { X } ) )  Fn  dom  ( G  |`  { X }
)  ->  Fun  ( F  o.  ( G  |`  { X } ) ) )
2422, 23syl 16 . 2  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  o.  ( G  |`  { X }
) ) )
25 resco 5316 . . 3  |-  ( ( F  o.  G )  |`  { X } )  =  ( F  o.  ( G  |`  { X } ) )
2625funeqi 5416 . 2  |-  ( Fun  ( ( F  o.  G )  |`  { X } )  <->  Fun  ( F  o.  ( G  |`  { X } ) ) )
2724, 26sylibr 204 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 359    = wceq 1649    e. wcel 1717   {csn 3759   dom cdm 4820   ran crn 4821    |` cres 4822   "cima 4823    o. ccom 4824   Fun wfun 5390    Fn wfn 5391   ` cfv 5396
This theorem is referenced by:  afvco2  27711
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404
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