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Theorem f1ococnv1 5509
Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
Assertion
Ref Expression
f1ococnv1  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )

Proof of Theorem f1ococnv1
StepHypRef Expression
1 f1orel 5483 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Rel  F )
2 dfrel2 5097 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
31, 2sylib 122 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' `' F  =  F )
43coeq2d 4807 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  `' `' F )  =  ( `' F  o.  F
) )
5 f1ocnv 5493 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
6 f1ococnv2 5507 . . 3  |-  ( `' F : B -1-1-onto-> A  -> 
( `' F  o.  `' `' F )  =  (  _I  |`  A )
)
75, 6syl 14 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  `' `' F )  =  (  _I  |`  A )
)
84, 7eqtr3d 2224 1  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    _I cid 4306   `'ccnv 4643    |` cres 4646    o. ccom 4648   Rel wrel 4649   -1-1-onto->wf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242
This theorem is referenced by:  f1cocnv1  5510  f1ocnvfv1  5799  fcof1o  5811  mapen  6874  hashfacen  10848
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