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Theorem f1cocnv2 5517
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 5455 . 2  |-  ( F : A -1-1-> B  ->  Fun  F )
2 funcocnv2 5514 . 2  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
31, 2syl 15 1  |-  ( F : A -1-1-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    _I cid 4320   `'ccnv 4704   ran crn 4706    |` cres 4707    o. ccom 4709   Fun wfun 5265   -1-1->wf1 5268
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276
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