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Theorem fococnv2 5486
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 5438 . . 3  |-  ( F : A -onto-> B  ->  Fun  F )
2 funcocnv2 5485 . . 3  |-  ( Fun 
F  ->  ( F  o.  `' F )  =  (  _I  |`  ran  F ) )
31, 2syl 14 . 2  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  ran  F ) )
4 forn 5440 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
54reseq2d 4906 . 2  |-  ( F : A -onto-> B  -> 
(  _I  |`  ran  F
)  =  (  _I  |`  B ) )
63, 5eqtrd 2210 1  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    _I cid 4287   `'ccnv 4624   ran crn 4626    |` cres 4627    o. ccom 4629   Fun wfun 5209   -onto->wfo 5213
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-fun 5217  df-fn 5218  df-f 5219  df-fo 5221
This theorem is referenced by:  f1ococnv2  5487  foeqcnvco  5788
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