ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fcof1o Unicode version

Theorem fcof1o 5457
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 5451 . . . 4  |-  ( ( F : A --> B  /\  ( G  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
21ad2ant2rl 488 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-> B )
3 fcofo 5452 . . . . 5  |-  ( ( F : A --> B  /\  G : B --> A  /\  ( F  o.  G
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
433expa 1115 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  ( F  o.  G )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
54adantrr 456 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -onto-> B )
6 df-f1o 4937 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
72, 5, 6sylanbrc 402 . 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 491 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
98coeq2d 4526 . . 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 4867 . . . 4  |-  ( ( `' F  o.  F
)  o.  G )  =  ( `' F  o.  ( F  o.  G
) )
11 f1ococnv1 5183 . . . . . . 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 4525 . . . . 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 5099 . . . . . 6  |-  ( G : B --> A  -> 
( (  _I  |`  A )  o.  G )  =  G )
1514ad2antlr 466 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
(  _I  |`  A )  o.  G )  =  G )
1613, 15eqtrd 2088 . . . 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 2102 . . 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 5167 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
19 f1of 5154 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
20 fcoi1 5098 . . . 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 2097 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
237, 22jca 294 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 101    = wceq 1259    _I cid 4053   `'ccnv 4372    |` cres 4375    o. ccom 4377   -->wf 4926   -1-1->wf1 4927   -onto->wfo 4928   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator