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Theorem fthinv 13800
Description: A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthsect.b  |-  B  =  ( Base `  C
)
fthsect.h  |-  H  =  (  Hom  `  C
)
fthsect.f  |-  ( ph  ->  F ( C Faith  D
) G )
fthsect.x  |-  ( ph  ->  X  e.  B )
fthsect.y  |-  ( ph  ->  Y  e.  B )
fthsect.m  |-  ( ph  ->  M  e.  ( X H Y ) )
fthsect.n  |-  ( ph  ->  N  e.  ( Y H X ) )
fthinv.s  |-  I  =  (Inv `  C )
fthinv.t  |-  J  =  (Inv `  D )
Assertion
Ref Expression
fthinv  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) ) )

Proof of Theorem fthinv
StepHypRef Expression
1 fthsect.b . . . 4  |-  B  =  ( Base `  C
)
2 fthsect.h . . . 4  |-  H  =  (  Hom  `  C
)
3 fthsect.f . . . 4  |-  ( ph  ->  F ( C Faith  D
) G )
4 fthsect.x . . . 4  |-  ( ph  ->  X  e.  B )
5 fthsect.y . . . 4  |-  ( ph  ->  Y  e.  B )
6 fthsect.m . . . 4  |-  ( ph  ->  M  e.  ( X H Y ) )
7 fthsect.n . . . 4  |-  ( ph  ->  N  e.  ( Y H X ) )
8 eqid 2283 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
9 eqid 2283 . . . 4  |-  (Sect `  D )  =  (Sect `  D )
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 13799 . . 3  |-  ( ph  ->  ( M ( X (Sect `  C ) Y ) N  <->  ( ( X G Y ) `  M ) ( ( F `  X ) (Sect `  D )
( F `  Y
) ) ( ( Y G X ) `
 N ) ) )
111, 2, 3, 5, 4, 7, 6, 8, 9fthsect 13799 . . 3  |-  ( ph  ->  ( N ( Y (Sect `  C ) X ) M  <->  ( ( Y G X ) `  N ) ( ( F `  Y ) (Sect `  D )
( F `  X
) ) ( ( X G Y ) `
 M ) ) )
1210, 11anbi12d 691 . 2  |-  ( ph  ->  ( ( M ( X (Sect `  C
) Y ) N  /\  N ( Y (Sect `  C ) X ) M )  <-> 
( ( ( X G Y ) `  M ) ( ( F `  X ) (Sect `  D )
( F `  Y
) ) ( ( Y G X ) `
 N )  /\  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  D ) ( F `
 X ) ) ( ( X G Y ) `  M
) ) ) )
13 fthinv.s . . 3  |-  I  =  (Inv `  C )
14 fthfunc 13781 . . . . . . . 8  |-  ( C Faith 
D )  C_  ( C  Func  D )
1514ssbri 4065 . . . . . . 7  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
163, 15syl 15 . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
17 df-br 4024 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1816, 17sylib 188 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
19 funcrcl 13737 . . . . 5  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
2018, 19syl 15 . . . 4  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2120simpld 445 . . 3  |-  ( ph  ->  C  e.  Cat )
221, 13, 21, 4, 5, 8isinv 13662 . 2  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  C ) Y ) N  /\  N ( Y (Sect `  C ) X ) M ) ) )
23 eqid 2283 . . 3  |-  ( Base `  D )  =  (
Base `  D )
24 fthinv.t . . 3  |-  J  =  (Inv `  D )
2520simprd 449 . . 3  |-  ( ph  ->  D  e.  Cat )
261, 23, 16funcf1 13740 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
2726, 4ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
2826, 5ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
2923, 24, 25, 27, 28, 9isinv 13662 . 2  |-  ( ph  ->  ( ( ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N )  <-> 
( ( ( X G Y ) `  M ) ( ( F `  X ) (Sect `  D )
( F `  Y
) ) ( ( Y G X ) `
 N )  /\  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  D ) ( F `
 X ) ) ( ( X G Y ) `  M
) ) ) )
3012, 22, 293bitr4d 276 1  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148    Hom chom 13219   Catccat 13566  Sectcsect 13647  Invcinv 13648    Func cfunc 13728   Faith cfth 13777
This theorem is referenced by:  ffthiso  13803
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  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  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-sect 13650  df-inv 13651  df-func 13732  df-fth 13779
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