Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cocnv Structured version   Unicode version

Theorem cocnv 28768
Description: Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
cocnv  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )

Proof of Theorem cocnv
StepHypRef Expression
1 coass 5465 . 2  |-  ( ( F  o.  G )  o.  `' G )  =  ( F  o.  ( G  o.  `' G ) )
2 funcocnv2 5774 . . . . 5  |-  ( Fun 
G  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
32adantl 466 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( G  o.  `' G )  =  (  _I  |`  ran  G ) )
43coeq2d 5111 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  o.  (  _I  |`  ran  G ) ) )
5 resco 5451 . . . 4  |-  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  o.  (  _I  |`  ran  G ) )
6 funrel 5544 . . . . . . 7  |-  ( Fun 
F  ->  Rel  F )
7 coi1 5462 . . . . . . 7  |-  ( Rel 
F  ->  ( F  o.  _I  )  =  F )
86, 7syl 16 . . . . . 6  |-  ( Fun 
F  ->  ( F  o.  _I  )  =  F )
98reseq1d 5218 . . . . 5  |-  ( Fun 
F  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
109adantr 465 . . . 4  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  _I  )  |` 
ran  G )  =  ( F  |`  ran  G
) )
115, 10syl5eqr 2509 . . 3  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  (  _I  |`  ran  G
) )  =  ( F  |`  ran  G ) )
124, 11eqtrd 2495 . 2  |-  ( ( Fun  F  /\  Fun  G )  ->  ( F  o.  ( G  o.  `' G ) )  =  ( F  |`  ran  G
) )
131, 12syl5eq 2507 1  |-  ( ( Fun  F  /\  Fun  G )  ->  ( ( F  o.  G )  o.  `' G )  =  ( F  |`  ran  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    _I cid 4740   `'ccnv 4948   ran crn 4950    |` cres 4951    o. ccom 4953   Rel wrel 4954   Fun wfun 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-fun 5529
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator