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Theorem f1ococnv2 5840
Description: The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
f1ococnv2  |-  ( F : A -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)

Proof of Theorem f1ococnv2
StepHypRef Expression
1 f1ofo 5821 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2 fococnv2 5839 . 2  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
31, 2syl 16 1  |-  ( F : A -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    _I cid 4790   `'ccnv 4998    |` cres 5001    o. ccom 5003   -onto->wfo 5584   -1-1-onto->wf1o 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593
This theorem is referenced by:  f1ococnv1  5842  f1ocnvfv2  6169  mapen  7678  hashfacen  12463  setcinv  15268  catcisolem  15284  symginv  16219  f1omvdco2  16266  gsumval3OLD  16696  gsumval3  16699  gsumzf1o  16705  gsumzf1oOLD  16708  psrass1lem  17797  evl1var  18140  pf1ind  18159  fcobij  27217  erdsze2lem2  28285  eldioph2  30297  ltrncoidN  34924  cdlemg46  35531  cdlemk45  35743  cdlemk55a  35755  tendocnv  35818
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