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Theorem fcof1o 5582
Description: Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
fcof1o  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F : A -1-1-onto-> B  /\  `' F  =  G ) )

Proof of Theorem fcof1o
StepHypRef Expression
1 fcof1 5576 . . . 4  |-  ( ( F : A --> B  /\  ( G  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
21ad2ant2rl 496 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-> B )
3 fcofo 5577 . . . . 5  |-  ( ( F : A --> B  /\  G : B --> A  /\  ( F  o.  G
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
433expa 1144 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  ( F  o.  G )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
54adantrr 464 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -onto-> B )
6 df-f1o 5035 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
72, 5, 6sylanbrc 409 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-onto-> B )
8 simprl 499 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
98coeq2d 4611 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  ( F  o.  G )
)  =  ( `' F  o.  (  _I  |`  B ) ) )
10 coass 4962 . . . 4  |-  ( ( `' F  o.  F
)  o.  G )  =  ( `' F  o.  ( F  o.  G
) )
11 f1ococnv1 5295 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
127, 11syl 14 . . . . . 6  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  F
)  =  (  _I  |`  A ) )
1312coeq1d 4610 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
( `' F  o.  F )  o.  G
)  =  ( (  _I  |`  A )  o.  G ) )
14 fcoi2 5205 . . . . . 6  |-  ( G : B --> A  -> 
( (  _I  |`  A )  o.  G )  =  G )
1514ad2antlr 474 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
(  _I  |`  A )  o.  G )  =  G )
1613, 15eqtrd 2121 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
( `' F  o.  F )  o.  G
)  =  G )
1710, 16syl5eqr 2135 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  ( F  o.  G )
)  =  G )
18 f1ocnv 5279 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
19 f1of 5266 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
20 fcoi1 5204 . . . 4  |-  ( `' F : B --> A  -> 
( `' F  o.  (  _I  |`  B ) )  =  `' F
)
217, 18, 19, 204syl 18 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  (  _I  |`  B ) )  =  `' F )
229, 17, 213eqtr3rd 2130 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
237, 22jca 301 1  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F : A -1-1-onto-> B  /\  `' F  =  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1290    _I cid 4124   `'ccnv 4451    |` cres 4454    o. ccom 4456   -->wf 5024   -1-1->wf1 5025   -onto->wfo 5026   -1-1-onto->wf1o 5027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036
This theorem is referenced by: (None)
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