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Theorem f1ococnv1 5207
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 5181 . . . 4  |-  ( F : A -1-1-onto-> B  ->  Rel  F )
2 dfrel2 4821 . . . 4  |-  ( Rel 
F  <->  `' `' F  =  F
)
31, 2sylib 120 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' `' F  =  F )
43coeq2d 4546 . 2  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  `' `' F )  =  ( `' F  o.  F
) )
5 f1ocnv 5191 . . 3  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
6 f1ococnv2 5205 . . 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 2117 1  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    _I cid 4071   `'ccnv 4390    |` cres 4393    o. ccom 4395   Rel wrel 4396   -1-1-onto->wf1o 4951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959
This theorem is referenced by:  f1cocnv1  5208  f1ocnvfv1  5469  fcof1o  5481
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