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Theorem fococnv2 5661
Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
Assertion
Ref Expression
fococnv2  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)

Proof of Theorem fococnv2
StepHypRef Expression
1 fofun 5616 . . 3  |-  ( F : A -onto-> B  ->  Fun  F )
2 funcocnv2 5660 . . 3  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
31, 2syl 16 . 2  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )
4 forn 5618 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
54reseq2d 5105 . 2  |-  ( F : A -onto-> B  -> 
(  _I  |`  ran  F
)  =  (  _I  |`  B ) )
63, 5eqtrd 2470 1  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    _I cid 4626   `'ccnv 4834   ran crn 4836    |` cres 4837    o. ccom 4839   Fun wfun 5407   -onto->wfo 5411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-fun 5415  df-fn 5416  df-f 5417  df-fo 5419
This theorem is referenced by:  f1ococnv2  5662  foeqcnvco  5993
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