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Theorem funcoressn 30171
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 30166 . . . . . . . 8  |-  ( ( G `  X )  e.  dom  F  ->  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } )
2 df-fn 5519 . . . . . . . . 9  |-  ( ( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) }  <->  ( Fun  ( F  |`  { ( G `  X ) } )  /\  dom  ( F  |`  { ( G `  X ) } )  =  {
( G `  X
) } ) )
32simplbi2com 627 . . . . . . . 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 429 . . . . . 6  |-  ( ( ( G `  X
)  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  ->  ( F  |` 
{ ( G `  X ) } )  Fn  { ( G `
 X ) } )
65adantr 465 . . . . 5  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( F  |`  { ( G `  X ) } )  Fn  {
( G `  X
) } )
7 fnsnfv 5850 . . . . . . . . 9  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  { ( G `  X ) }  =  ( G " { X } ) )
87adantl 466 . . . . . . . 8  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ( G " { X } ) )
9 df-ima 4951 . . . . . . . 8  |-  ( G
" { X }
)  =  ran  ( G  |`  { X }
)
108, 9syl6eq 2508 . . . . . . 7  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  { ( G `  X ) }  =  ran  ( G  |`  { X } ) )
1110reseq2d 5208 . . . . . 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 5601 . . . . 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 210 . . . 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 5606 . . . . . . 7  |-  ( G  Fn  A  ->  Fun  G )
15 funres 5555 . . . . . . . 8  |-  ( Fun 
G  ->  Fun  ( G  |`  { X } ) )
16 funfn 5545 . . . . . . . 8  |-  ( Fun  ( G  |`  { X } )  <->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
1715, 16sylib 196 . . . . . . 7  |-  ( Fun 
G  ->  ( G  |` 
{ X } )  Fn  dom  ( G  |`  { X } ) )
1814, 17syl 16 . . . . . 6  |-  ( G  Fn  A  ->  ( G  |`  { X }
)  Fn  dom  ( G  |`  { X }
) )
1918adantr 465 . . . . 5  |-  ( ( G  Fn  A  /\  X  e.  A )  ->  ( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
2019adantl 466 . . . 4  |-  ( ( ( ( G `  X )  e.  dom  F  /\  Fun  ( F  |`  { ( G `  X ) } ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  -> 
( G  |`  { X } )  Fn  dom  ( G  |`  { X } ) )
21 fnresfnco 30170 . . . 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 661 . . 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 5606 . . 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 5440 . . 3  |-  ( ( F  o.  G )  |`  { X } )  =  ( F  o.  ( G  |`  { X } ) )
2625funeqi 5536 . 2  |-  ( Fun  ( ( F  o.  G )  |`  { X } )  <->  Fun  ( F  o.  ( G  |`  { X } ) ) )
2724, 26sylibr 212 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3975   dom cdm 4938   ran crn 4939    |` cres 4940   "cima 4941    o. ccom 4942   Fun wfun 5510    Fn wfn 5511   ` cfv 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-fv 5524
This theorem is referenced by:  afvco2  30220
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