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Theorem funresfunco 27956
Description: Composition of two functions, generalization of funco 5483. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
funresfunco  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )

Proof of Theorem funresfunco
StepHypRef Expression
1 funco 5483 . 2  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( ( F  |`  ran  G )  o.  G ) )
2 ssid 3359 . . . . 5  |-  ran  G  C_ 
ran  G
3 cores 5365 . . . . 5  |-  ( ran 
G  C_  ran  G  -> 
( ( F  |`  ran  G )  o.  G
)  =  ( F  o.  G ) )
42, 3ax-mp 8 . . . 4  |-  ( ( F  |`  ran  G )  o.  G )  =  ( F  o.  G
)
54eqcomi 2439 . . 3  |-  ( F  o.  G )  =  ( ( F  |`  ran  G )  o.  G
)
65funeqi 5466 . 2  |-  ( Fun  ( F  o.  G
)  <->  Fun  ( ( F  |`  ran  G )  o.  G ) )
71, 6sylibr 204 1  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    C_ wss 3312   ran crn 4871    |` cres 4872    o. ccom 4874   Fun wfun 5440
This theorem is referenced by:  fnresfnco  27957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-fun 5448
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