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

Theorem fcof1o 5658
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 5652 . . . 4  |-  ( ( F : A --> B  /\  ( G  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
21ad2ant2rl 502 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-> B )
3 fcofo 5653 . . . . 5  |-  ( ( F : A --> B  /\  G : B --> A  /\  ( F  o.  G
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
433expa 1166 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  ( F  o.  G )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
54adantrr 470 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -onto-> B )
6 df-f1o 5100 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
72, 5, 6sylanbrc 413 . 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 505 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
98coeq2d 4671 . . 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 5027 . . . 4  |-  ( ( `' F  o.  F
)  o.  G )  =  ( `' F  o.  ( F  o.  G
) )
11 f1ococnv1 5364 . . . . . . 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 4670 . . . . 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 5274 . . . . . 6  |-  ( G : B --> A  -> 
( (  _I  |`  A )  o.  G )  =  G )
1514ad2antlr 480 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
(  _I  |`  A )  o.  G )  =  G )
1613, 15eqtrd 2150 . . . 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 2164 . . 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 5348 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
19 f1of 5335 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
20 fcoi1 5273 . . . 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 2159 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
237, 22jca 304 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 1316    _I cid 4180   `'ccnv 4508    |` cres 4511    o. ccom 4513   -->wf 5089   -1-1->wf1 5090   -onto->wfo 5091   -1-1-onto->wf1o 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  txswaphmeo  12401
  Copyright terms: Public domain W3C validator