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Theorem fthinv 15417
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 2454 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
9 eqid 2454 . . . 4  |-  (Sect `  D )  =  (Sect `  D )
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 15416 . . 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 15416 . . 3  |-  ( ph  ->  ( N ( Y (Sect `  C ) X ) M  <->  ( ( Y G X ) `  N ) ( ( F `  Y ) (Sect `  D )
( F `  X
) ) ( ( X G Y ) `
 M ) ) )
1210, 11anbi12d 708 . 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 15398 . . . . . . . 8  |-  ( C Faith 
D )  C_  ( C  Func  D )
1514ssbri 4481 . . . . . . 7  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
163, 15syl 16 . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
17 df-br 4440 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1816, 17sylib 196 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
19 funcrcl 15354 . . . . 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 457 . . 3  |-  ( ph  ->  C  e.  Cat )
221, 13, 21, 4, 5, 8isinv 15251 . 2  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  C ) Y ) N  /\  N ( Y (Sect `  C ) X ) M ) ) )
23 eqid 2454 . . 3  |-  ( Base `  D )  =  (
Base `  D )
24 fthinv.t . . 3  |-  J  =  (Inv `  D )
2520simprd 461 . . 3  |-  ( ph  ->  D  e.  Cat )
261, 23, 16funcf1 15357 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
2726, 4ffvelrnd 6008 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
2826, 5ffvelrnd 6008 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
2923, 24, 25, 27, 28, 9isinv 15251 . 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 285 1  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14719   Hom chom 14798   Catccat 15156  Sectcsect 15235  Invcinv 15236    Func cfunc 15345   Faith cfth 15394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-ixp 7463  df-cat 15160  df-cid 15161  df-sect 15238  df-inv 15239  df-func 15349  df-fth 15396
This theorem is referenced by:  ffthiso  15420
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