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Theorem f1ococnv1 5521
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 5495 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Rel  F )
2 dfrel2 5108 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
31, 2sylib 122 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' `' F  =  F )
43coeq2d 4818 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  `' `' F )  =  ( `' F  o.  F
) )
5 f1ocnv 5505 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
6 f1ococnv2 5519 . . 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 2228 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 4317   `'ccnv 4654    |` cres 4657    o. ccom 4659   Rel wrel 4660   -1-1-onto->wf1o 5245
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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253
This theorem is referenced by:  f1cocnv1  5522  f1ocnvfv1  5812  fcof1o  5824  mapen  6893  hashfacen  10897
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