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Theorem f1cocnv2 5782
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv2  |-  ( F : A -1-1-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )

Proof of Theorem f1cocnv2
StepHypRef Expression
1 f1fun 5722 . 2  |-  ( F : A -1-1-> B  ->  Fun  F )
2 funcocnv2 5779 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
31, 2syl 17 1  |-  ( F : A -1-1-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    _I cid 4732   `'ccnv 4941   ran crn 4943    |` cres 4944    o. ccom 4946   Fun wfun 5519   -1-1->wf1 5522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530
This theorem is referenced by: (None)
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