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Theorem f1ococnv2 5362
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 5342 . 2  |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
2 fococnv2 5361 . 2  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
31, 2syl 14 1  |-  ( F : A -1-1-onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316    _I cid 4180   `'ccnv 4508    |` cres 4511    o. ccom 4513   -onto->wfo 5091   -1-1-onto->wf1o 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100
This theorem is referenced by:  f1ococnv1  5364  f1ocnvfv2  5647  mapen  6708  hashfacen  10547
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