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Theorem f1cocnv2 5775
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 5715 . 2  |-  ( F : A -1-1-> B  ->  Fun  F )
2 funcocnv2 5772 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
31, 2syl 16 1  |-  ( F : A -1-1-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    _I cid 4738   `'ccnv 4946   ran crn 4948    |` cres 4949    o. ccom 4951   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 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 4520  ax-nul 4528  ax-pr 4638
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 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530
This theorem is referenced by: (None)
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