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Theorem f1ococnv2 5662
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 5643 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2 fococnv2 5661 . 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 1369    _I cid 4626   `'ccnv 4834    |` cres 4837    o. ccom 4839   -onto->wfo 5411   -1-1-onto->wf1o 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420
This theorem is referenced by:  f1ococnv1  5664  f1ocnvfv2  5979  mapen  7467  hashfacen  12199  setcinv  14950  catcisolem  14966  symginv  15898  f1omvdco2  15945  gsumval3OLD  16373  gsumval3  16376  gsumzf1o  16382  gsumzf1oOLD  16385  psrass1lem  17424  evl1var  17745  pf1ind  17764  fcobij  25976  erdsze2lem2  27044  eldioph2  29053  ltrncoidN  33612  cdlemg46  34219  cdlemk45  34431  cdlemk55a  34443  tendocnv  34506
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