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Theorem f1ococnv2 5824
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 5805 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2 fococnv2 5823 . 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 1398    _I cid 4779   `'ccnv 4987    |` cres 4990    o. ccom 4992   -onto->wfo 5568   -1-1-onto->wf1o 5569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577
This theorem is referenced by:  f1ococnv1  5826  f1ocnvfv2  6158  mapen  7674  hashfacen  12487  setcinv  15568  catcisolem  15584  symginv  16626  f1omvdco2  16672  gsumval3OLD  17107  gsumval3  17110  gsumzf1o  17116  gsumzf1oOLD  17119  psrass1lem  18224  evl1var  18567  pf1ind  18586  fcobij  27779  erdsze2lem2  28912  eldioph2  30934  rngcinv  33043  rngcinvALTV  33055  ringcinv  33094  ringcinvALTV  33118  ltrncoidN  36249  cdlemg46  36858  cdlemk45  37070  cdlemk55a  37082  tendocnv  37145
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