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Theorem fnresfnco 27980
Description: Composition of two functions, similar to fnco 5556. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
fnresfnco  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ( F  o.  G )  Fn  B
)

Proof of Theorem fnresfnco
StepHypRef Expression
1 fnfun 5545 . . 3  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  Fun  ( F  |`  ran  G
) )
2 fnfun 5545 . . 3  |-  ( G  Fn  B  ->  Fun  G )
3 funresfunco 27979 . . 3  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 465 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  Fun  ( F  o.  G ) )
5 fndm 5547 . . . . . 6  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  dom  ( F  |`  ran  G
)  =  ran  G
)
6 dmres 5170 . . . . . . . 8  |-  dom  ( F  |`  ran  G )  =  ( ran  G  i^i  dom  F )
76eqeq1i 2445 . . . . . . 7  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  <->  ( ran  G  i^i  dom  F )  =  ran  G
)
8 df-ss 3336 . . . . . . . 8  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
98biimpri 199 . . . . . . 7  |-  ( ( ran  G  i^i  dom  F )  =  ran  G  ->  ran  G  C_  dom  F )
107, 9sylbi 189 . . . . . 6  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  ->  ran  G  C_  dom  F )
115, 10syl 16 . . . . 5  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  ran  G  C_  dom  F )
1211adantr 453 . . . 4  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ran  G  C_  dom  F )
13 dmcosseq 5140 . . . 4  |-  ( ran 
G  C_  dom  F  ->  dom  ( F  o.  G
)  =  dom  G
)
1412, 13syl 16 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  dom  G )
15 fndm 5547 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
1615adantl 454 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  G  =  B )
1714, 16eqtrd 2470 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  B )
18 df-fn 5460 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
194, 17, 18sylanbrc 647 1  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    i^i cin 3321    C_ wss 3322   dom cdm 4881   ran crn 4882    |` cres 4883    o. ccom 4885   Fun wfun 5451    Fn wfn 5452
This theorem is referenced by:  funcoressn  27981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-fun 5459  df-fn 5460
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