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Theorem f1ococnv2 5642
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 5621 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2 fococnv2 5641 . 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 set class
Syntax hints:    -> wi 4    = wceq 1649    _I cid 4434   `'ccnv 4817    |` cres 4820    o. ccom 4822   -onto->wfo 5392   -1-1-onto->wf1o 5393
This theorem is referenced by:  f1ococnv1  5644  f1ocnvfv2  5954  mapen  7207  hashfacen  11630  setcinv  14172  catcisolem  14188  symginv  15032  gsumval3  15441  gsumzf1o  15446  psrass1lem  16369  evl1var  19819  pf1ind  19842  erdsze2lem2  24669  eldioph2  26511  f1omvdco2  27060  ltrncoidN  30242  cdlemg46  30849  cdlemk45  31061  cdlemk55a  31073  tendocnv  31136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401
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