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Theorem fcof1o 5803
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 5797 . . . 4  |-  ( ( F : A --> B  /\  ( G  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
21ad2ant2rl 729 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-> B )
3 fcofo 5798 . . . . 5  |-  ( ( F : A --> B  /\  G : B --> A  /\  ( F  o.  G
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
433expa 1151 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  ( F  o.  G )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
54adantrr 697 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -onto-> B )
6 df-f1o 5262 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
72, 5, 6sylanbrc 645 . 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 732 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
98coeq2d 4846 . . 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 5191 . . . 4  |-  ( ( `' F  o.  F
)  o.  G )  =  ( `' F  o.  ( F  o.  G
) )
11 f1ococnv1 5502 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
127, 11syl 15 . . . . . 6  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  F
)  =  (  _I  |`  A ) )
1312coeq1d 4845 . . . . 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 5416 . . . . . 6  |-  ( G : B --> A  -> 
( (  _I  |`  A )  o.  G )  =  G )
1514ad2antlr 707 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
(  _I  |`  A )  o.  G )  =  G )
1613, 15eqtrd 2315 . . . 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 2329 . . 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 5485 . . . . 5  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
19 f1of 5472 . . . . 5  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
207, 18, 193syl 18 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F : B --> A )
21 fcoi1 5415 . . . 4  |-  ( `' F : B --> A  -> 
( `' F  o.  (  _I  |`  B ) )  =  `' F
)
2220, 21syl 15 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  (  _I  |`  B ) )  =  `' F )
239, 17, 223eqtr3rd 2324 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
247, 23jca 518 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 358    = wceq 1623    _I cid 4304   `'ccnv 4688    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  setcinv  13922  catciso  13939  yonedainv  14055  txswaphmeo  17496  qtophmeo  17508  pmtrff1o  27404  pmtrfcnv  27405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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