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Theorem fcof1o 6109
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 6103 . . . 4  |-  ( ( F : A --> B  /\  ( G  o.  F
)  =  (  _I  |`  A ) )  ->  F : A -1-1-> B )
21ad2ant2rl 748 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -1-1-> B )
3 fcofo 6104 . . . . 5  |-  ( ( F : A --> B  /\  G : B --> A  /\  ( F  o.  G
)  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
433expa 1188 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  ( F  o.  G )  =  (  _I  |`  B ) )  ->  F : A -onto-> B )
54adantrr 716 . . 3  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  F : A -onto-> B )
6 df-f1o 5536 . . 3  |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  F : A -onto-> B ) )
72, 5, 6sylanbrc 664 . 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 755 . . . 4  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( F  o.  G )  =  (  _I  |`  B ) )
98coeq2d 5113 . . 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 5467 . . . 4  |-  ( ( `' F  o.  F
)  o.  G )  =  ( `' F  o.  ( F  o.  G
) )
11 f1ococnv1 5780 . . . . . . 7  |-  ( F : A -1-1-onto-> B  ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
127, 11syl 16 . . . . . 6  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  ( `' F  o.  F
)  =  (  _I  |`  A ) )
1312coeq1d 5112 . . . . 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 5697 . . . . . 6  |-  ( G : B --> A  -> 
( (  _I  |`  A )  o.  G )  =  G )
1514ad2antlr 726 . . . . 5  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  (
(  _I  |`  A )  o.  G )  =  G )
1613, 15eqtrd 2495 . . . 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 2509 . . 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 5764 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
19 f1of 5752 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B --> A )
20 fcoi1 5696 . . . 4  |-  ( `' F : B --> A  -> 
( `' F  o.  (  _I  |`  B ) )  =  `' F
)
217, 18, 19, 204syl 21 . . 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 2504 . 2  |-  ( ( ( F : A --> B  /\  G : B --> A )  /\  (
( F  o.  G
)  =  (  _I  |`  B )  /\  ( G  o.  F )  =  (  _I  |`  A ) ) )  ->  `' F  =  G )
237, 22jca 532 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    _I cid 4742   `'ccnv 4950    |` cres 4953    o. ccom 4955   -->wf 5525   -1-1->wf1 5526   -onto->wfo 5527   -1-1-onto->wf1o 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537
This theorem is referenced by:  setcinv  15080  catciso  15097  yonedainv  15213  pmtrff1o  16091  pmtrfcnv  16092  evpmodpmf1o  18154  txswaphmeo  19513  qtophmeo  19525  m2cpminv  31268
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