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Theorem fthsect 13799
Description: A faithful functor reflects sections. (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 ) )
fthsect.s  |-  S  =  (Sect `  C )
fthsect.t  |-  T  =  (Sect `  D )
Assertion
Ref Expression
fthsect  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) T ( F `  Y
) ) ( ( Y G X ) `
 N ) ) )

Proof of Theorem fthsect
StepHypRef Expression
1 fthsect.b . . . 4  |-  B  =  ( Base `  C
)
2 fthsect.h . . . 4  |-  H  =  (  Hom  `  C
)
3 eqid 2283 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 fthsect.f . . . 4  |-  ( ph  ->  F ( C Faith  D
) G )
5 fthsect.x . . . 4  |-  ( ph  ->  X  e.  B )
6 eqid 2283 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 fthfunc 13781 . . . . . . . . . 10  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4065 . . . . . . . . 9  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
94, 8syl 15 . . . . . . . 8  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4024 . . . . . . . 8  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 188 . . . . . . 7  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 13737 . . . . . . 7  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1311, 12syl 15 . . . . . 6  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1413simpld 445 . . . . 5  |-  ( ph  ->  C  e.  Cat )
15 fthsect.y . . . . 5  |-  ( ph  ->  Y  e.  B )
16 fthsect.m . . . . 5  |-  ( ph  ->  M  e.  ( X H Y ) )
17 fthsect.n . . . . 5  |-  ( ph  ->  N  e.  ( Y H X ) )
181, 2, 6, 14, 5, 15, 5, 16, 17catcocl 13587 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  C ) X ) M )  e.  ( X H X ) )
19 eqid 2283 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
201, 2, 19, 14, 5catidcl 13584 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
211, 2, 3, 4, 5, 5, 18, 20fthi 13792 . . 3  |-  ( ph  ->  ( ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  C
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  C ) `
 X ) )  <-> 
( N ( <. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )
) )
22 eqid 2283 . . . . 5  |-  (comp `  D )  =  (comp `  D )
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 13745 . . . 4  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  C
) X ) M ) )  =  ( ( ( Y G X ) `  N
) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  D ) ( F `
 X ) ) ( ( X G Y ) `  M
) ) )
24 eqid 2283 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
251, 19, 24, 9, 5funcid 13744 . . . 4  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  C
) `  X )
)  =  ( ( Id `  D ) `
 ( F `  X ) ) )
2623, 25eqeq12d 2297 . . 3  |-  ( ph  ->  ( ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  C
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  C ) `
 X ) )  <-> 
( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  D )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  D ) `  ( F `  X ) ) ) )
2721, 26bitr3d 246 . 2  |-  ( ph  ->  ( ( N (
<. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )  <->  ( ( ( Y G X ) `  N
) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  D ) ( F `
 X ) ) ( ( X G Y ) `  M
) )  =  ( ( Id `  D
) `  ( F `  X ) ) ) )
28 fthsect.s . . 3  |-  S  =  (Sect `  C )
291, 2, 6, 19, 28, 14, 5, 15, 16, 17issect2 13657 . 2  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( N ( <. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )
) )
30 eqid 2283 . . 3  |-  ( Base `  D )  =  (
Base `  D )
31 fthsect.t . . 3  |-  T  =  (Sect `  D )
3213simprd 449 . . 3  |-  ( ph  ->  D  e.  Cat )
331, 30, 9funcf1 13740 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
3433, 5ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
3533, 15ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
361, 2, 3, 9, 5, 15funcf2 13742 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) (  Hom  `  D
) ( F `  Y ) ) )
3736, 16ffvelrnd 5666 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  D
) ( F `  Y ) ) )
381, 2, 3, 9, 15, 5funcf2 13742 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y H X ) --> ( ( F `  Y
) (  Hom  `  D
) ( F `  X ) ) )
3938, 17ffvelrnd 5666 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  D
) ( F `  X ) ) )
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 13657 . 2  |-  ( ph  ->  ( ( ( X G Y ) `  M ) ( ( F `  X ) T ( F `  Y ) ) ( ( Y G X ) `  N )  <-> 
( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  D )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  D ) `  ( F `  X ) ) ) )
4127, 29, 403bitr4d 276 1  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) T ( 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  compcco 13220   Catccat 13566   Idccid 13567  Sectcsect 13647    Func cfunc 13728   Faith cfth 13777
This theorem is referenced by:  fthinv  13800
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-func 13732  df-fth 13779
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