Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  funresfunco Unicode version

Theorem funresfunco 27651
Description: Composition of two functions, generalization of funco 5424. (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 5424 . 2  |-  ( ( Fun  ( F  |`  ran  G )  /\  Fun  G )  ->  Fun  ( ( F  |`  ran  G )  o.  G ) )
2 ssid 3303 . . . . 5  |-  ran  G  C_ 
ran  G
3 cores 5306 . . . . 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 2384 . . 3  |-  ( F  o.  G )  =  ( ( F  |`  ran  G )  o.  G
)
65funeqi 5407 . 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 1649    C_ wss 3256   ran crn 4812    |` cres 4813    o. ccom 4815   Fun wfun 5381
This theorem is referenced by:  fnresfnco  27652
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-fun 5389
  Copyright terms: Public domain W3C validator