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Theorem fnresfnco 27314
Description: Composition of two functions, similar to fnco 5431. (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 5420 . . 3  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  Fun  ( F  |`  ran  G
) )
2 fnfun 5420 . . 3  |-  ( G  Fn  B  ->  Fun  G )
3 funresfunco 27313 . . 3  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 463 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  Fun  ( F  o.  G ) )
5 fndm 5422 . . . . . 6  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  dom  ( F  |`  ran  G
)  =  ran  G
)
6 dmres 5055 . . . . . . . 8  |-  dom  ( F  |`  ran  G )  =  ( ran  G  i^i  dom  F )
76eqeq1i 2365 . . . . . . 7  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  <->  ( ran  G  i^i  dom  F )  =  ran  G
)
8 df-ss 3242 . . . . . . . 8  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
98biimpri 197 . . . . . . 7  |-  ( ( ran  G  i^i  dom  F )  =  ran  G  ->  ran  G  C_  dom  F )
107, 9sylbi 187 . . . . . 6  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  ->  ran  G  C_  dom  F )
115, 10syl 15 . . . . 5  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  ran  G  C_  dom  F )
1211adantr 451 . . . 4  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ran  G  C_  dom  F )
13 dmcosseq 5025 . . . 4  |-  ( ran 
G  C_  dom  F  ->  dom  ( F  o.  G
)  =  dom  G
)
1412, 13syl 15 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  dom  G )
15 fndm 5422 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
1615adantl 452 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  G  =  B )
1714, 16eqtrd 2390 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  B )
18 df-fn 5337 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
194, 17, 18sylanbrc 645 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 358    = wceq 1642    i^i cin 3227    C_ wss 3228   dom cdm 4768   ran crn 4769    |` cres 4770    o. ccom 4772   Fun wfun 5328    Fn wfn 5329
This theorem is referenced by:  funcoressn  27315
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-fun 5336  df-fn 5337
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