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Theorem fnresfnco 31635
Description: Composition of two functions, similar to fnco 5682. (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 5671 . . 3  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  Fun  ( F  |`  ran  G
) )
2 fnfun 5671 . . 3  |-  ( G  Fn  B  ->  Fun  G )
3 funresfunco 31634 . . 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 5673 . . . . . 6  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  dom  ( F  |`  ran  G
)  =  ran  G
)
6 dmres 5287 . . . . . . . 8  |-  dom  ( F  |`  ran  G )  =  ( ran  G  i^i  dom  F )
76eqeq1i 2469 . . . . . . 7  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  <->  ( ran  G  i^i  dom  F )  =  ran  G
)
8 df-ss 3485 . . . . . . . 8  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
98biimpri 206 . . . . . . 7  |-  ( ( ran  G  i^i  dom  F )  =  ran  G  ->  ran  G  C_  dom  F )
107, 9sylbi 195 . . . . . 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 465 . . . 4  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ran  G  C_  dom  F )
13 dmcosseq 5257 . . . 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 5673 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
1615adantl 466 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  G  =  B )
1714, 16eqtrd 2503 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  B )
18 df-fn 5584 . 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 1374    i^i cin 3470    C_ wss 3471   dom cdm 4994   ran crn 4995    |` cres 4996    o. ccom 4998   Fun wfun 5575    Fn wfn 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-fun 5583  df-fn 5584
This theorem is referenced by:  funcoressn  31636
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