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Theorem fthinv 14123
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 2436 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
9 eqid 2436 . . . 4  |-  (Sect `  D )  =  (Sect `  D )
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 14122 . . 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 14122 . . 3  |-  ( ph  ->  ( N ( Y (Sect `  C ) X ) M  <->  ( ( Y G X ) `  N ) ( ( F `  Y ) (Sect `  D )
( F `  X
) ) ( ( X G Y ) `
 M ) ) )
1210, 11anbi12d 692 . 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 14104 . . . . . . . 8  |-  ( C Faith 
D )  C_  ( C  Func  D )
1514ssbri 4254 . . . . . . 7  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
163, 15syl 16 . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
17 df-br 4213 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1816, 17sylib 189 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
19 funcrcl 14060 . . . . 5  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
2018, 19syl 16 . . . 4  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
2120simpld 446 . . 3  |-  ( ph  ->  C  e.  Cat )
221, 13, 21, 4, 5, 8isinv 13985 . 2  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  C ) Y ) N  /\  N ( Y (Sect `  C ) X ) M ) ) )
23 eqid 2436 . . 3  |-  ( Base `  D )  =  (
Base `  D )
24 fthinv.t . . 3  |-  J  =  (Inv `  D )
2520simprd 450 . . 3  |-  ( ph  ->  D  e.  Cat )
261, 23, 16funcf1 14063 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
2726, 4ffvelrnd 5871 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
2826, 5ffvelrnd 5871 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
2923, 24, 25, 27, 28, 9isinv 13985 . 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 277 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469    Hom chom 13540   Catccat 13889  Sectcsect 13970  Invcinv 13971    Func cfunc 14051   Faith cfth 14100
This theorem is referenced by:  ffthiso  14126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-map 7020  df-ixp 7064  df-cat 13893  df-cid 13894  df-sect 13973  df-inv 13974  df-func 14055  df-fth 14102
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