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Theorem fnresfnco 27861
Description: Composition of two functions, similar to fnco 5516. (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 5505 . . 3  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  Fun  ( F  |`  ran  G
) )
2 fnfun 5505 . . 3  |-  ( G  Fn  B  ->  Fun  G )
3 funresfunco 27860 . . 3  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
41, 2, 3syl2an 464 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  Fun  ( F  o.  G ) )
5 fndm 5507 . . . . . 6  |-  ( ( F  |`  ran  G )  Fn  ran  G  ->  dom  ( F  |`  ran  G
)  =  ran  G
)
6 dmres 5130 . . . . . . . 8  |-  dom  ( F  |`  ran  G )  =  ( ran  G  i^i  dom  F )
76eqeq1i 2415 . . . . . . 7  |-  ( dom  ( F  |`  ran  G
)  =  ran  G  <->  ( ran  G  i^i  dom  F )  =  ran  G
)
8 df-ss 3298 . . . . . . . 8  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
98biimpri 198 . . . . . . 7  |-  ( ( ran  G  i^i  dom  F )  =  ran  G  ->  ran  G  C_  dom  F )
107, 9sylbi 188 . . . . . 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 452 . . . 4  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  ran  G  C_  dom  F )
13 dmcosseq 5100 . . . 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 5507 . . . 4  |-  ( G  Fn  B  ->  dom  G  =  B )
1615adantl 453 . . 3  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  G  =  B )
1714, 16eqtrd 2440 . 2  |-  ( ( ( F  |`  ran  G
)  Fn  ran  G  /\  G  Fn  B
)  ->  dom  ( F  o.  G )  =  B )
18 df-fn 5420 . 2  |-  ( ( F  o.  G )  Fn  B  <->  ( Fun  ( F  o.  G
)  /\  dom  ( F  o.  G )  =  B ) )
194, 17, 18sylanbrc 646 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 359    = wceq 1649    i^i cin 3283    C_ wss 3284   dom cdm 4841   ran crn 4842    |` cres 4843    o. ccom 4845   Fun wfun 5411    Fn wfn 5412
This theorem is referenced by:  funcoressn  27862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-fun 5419  df-fn 5420
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