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Theorem fthsect 14042
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 2380 . . . 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 2380 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 fthfunc 14024 . . . . . . . . . 10  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4188 . . . . . . . . 9  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
94, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4147 . . . . . . . 8  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 189 . . . . . . 7  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 13980 . . . . . . 7  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1311, 12syl 16 . . . . . 6  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1413simpld 446 . . . . 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 13830 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  C ) X ) M )  e.  ( X H X ) )
19 eqid 2380 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
201, 2, 19, 14, 5catidcl 13827 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
211, 2, 3, 4, 5, 5, 18, 20fthi 14035 . . 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 2380 . . . . 5  |-  (comp `  D )  =  (comp `  D )
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 13988 . . . 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 2380 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
251, 19, 24, 9, 5funcid 13987 . . . 4  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  C
) `  X )
)  =  ( ( Id `  D ) `
 ( F `  X ) ) )
2623, 25eqeq12d 2394 . . 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 247 . 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 13900 . 2  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( N ( <. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )
) )
30 eqid 2380 . . 3  |-  ( Base `  D )  =  (
Base `  D )
31 fthsect.t . . 3  |-  T  =  (Sect `  D )
3213simprd 450 . . 3  |-  ( ph  ->  D  e.  Cat )
331, 30, 9funcf1 13983 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
3433, 5ffvelrnd 5803 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
3533, 15ffvelrnd 5803 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
361, 2, 3, 9, 5, 15funcf2 13985 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) (  Hom  `  D
) ( F `  Y ) ) )
3736, 16ffvelrnd 5803 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  D
) ( F `  Y ) ) )
381, 2, 3, 9, 15, 5funcf2 13985 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y H X ) --> ( ( F `  Y
) (  Hom  `  D
) ( F `  X ) ) )
3938, 17ffvelrnd 5803 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  D
) ( F `  X ) ) )
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 13900 . 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 277 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389    Hom chom 13460  compcco 13461   Catccat 13809   Idccid 13810  Sectcsect 13890    Func cfunc 13971   Faith cfth 14020
This theorem is referenced by:  fthinv  14043
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-map 6949  df-ixp 6993  df-cat 13813  df-cid 13814  df-sect 13893  df-func 13975  df-fth 14022
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