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Theorem fnresfnco 37592
Description: Composition of two functions, similar to fnco 5672. (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 5661 . . 3  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  Fun  ( F  |`  ran  G
) )
2 fnfun 5661 . . 3  |-  ( G  Fn  B  ->  Fun  G )
3 funresfunco 37591 . . 3  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 477 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  Fun  ( F  o.  G ) )
5 fndm 5663 . . . . . 6  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  dom  ( F  |`  ran  G
)  =  ran  G
)
6 dmres 5116 . . . . . . . 8  |-  dom  ( F  |`  ran  G )  =  ( ran  G  i^i  dom  F )
76eqeq1i 2411 . . . . . . 7  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  <->  ( ran  G  i^i  dom  F )  =  ran  G
)
8 df-ss 3430 . . . . . . . 8  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
98biimpri 208 . . . . . . 7  |-  ( ( ran  G  i^i  dom  F )  =  ran  G  ->  ran  G  C_  dom  F )
107, 9sylbi 197 . . . . . 6  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  ->  ran  G  C_  dom  F )
115, 10syl 17 . . . . 5  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  ran  G  C_  dom  F )
1211adantr 465 . . . 4  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ran  G  C_  dom  F )
13 dmcosseq 5087 . . . 4  |-  ( ran 
G  C_  dom  F  ->  dom  ( F  o.  G
)  =  dom  G
)
1412, 13syl 17 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  dom  G )
15 fndm 5663 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
1615adantl 466 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  G  =  B )
1714, 16eqtrd 2445 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  B )
18 df-fn 5574 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
194, 17, 18sylanbrc 664 1  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ( F  o.  G )  Fn  B
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1407    i^i cin 3415    C_ wss 3416   dom cdm 4825   ran crn 4826    |` cres 4827    o. ccom 4829   Fun wfun 5565    Fn wfn 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-fun 5573  df-fn 5574
This theorem is referenced by:  funcoressn  37593
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