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Theorem fococnv2 5203
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 5158 . . 3  |-  ( F : A -onto-> B  ->  Fun  F )
2 funcocnv2 5202 . . 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 5160 . . 3  |-  ( F : A -onto-> B  ->  ran  F  =  B )
54reseq2d 4660 . 2  |-  ( F : A -onto-> B  -> 
(  _I  |`  ran  F
)  =  (  _I  |`  B ) )
63, 5eqtrd 2115 1  |-  ( F : A -onto-> B  -> 
( F  o.  `' F )  =  (  _I  |`  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    _I cid 4071   `'ccnv 4390   ran crn 4392    |` cres 4393    o. ccom 4395   Fun wfun 4946   -onto->wfo 4950
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-fo 4958
This theorem is referenced by:  f1ococnv2  5204  foeqcnvco  5481
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