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Theorem fococnv2 5839
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 5794 . . 3  |-  ( F : A -onto-> B  ->  Fun  F )
2 funcocnv2 5838 . . 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 5796 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
54reseq2d 5271 . 2  |-  ( F : A -onto-> B  -> 
(  _I  |`  ran  F
)  =  (  _I  |`  B ) )
63, 5eqtrd 2508 1  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    _I cid 4790   `'ccnv 4998   ran crn 5000    |` cres 5001    o. ccom 5003   Fun wfun 5580   -onto->wfo 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592
This theorem is referenced by:  f1ococnv2  5840  foeqcnvco  6189
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