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Theorem fthsect 14840
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 2443 . . . 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 2443 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 fthfunc 14822 . . . . . . . . . 10  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4339 . . . . . . . . 9  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
94, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4298 . . . . . . . 8  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
119, 10sylib 196 . . . . . . 7  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
12 funcrcl 14778 . . . . . . 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 459 . . . . 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 14628 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  C ) X ) M )  e.  ( X H X ) )
19 eqid 2443 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
201, 2, 19, 14, 5catidcl 14625 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
211, 2, 3, 4, 5, 5, 18, 20fthi 14833 . . 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 2443 . . . . 5  |-  (comp `  D )  =  (comp `  D )
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 14786 . . . 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 2443 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
251, 19, 24, 9, 5funcid 14785 . . . 4  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  C
) `  X )
)  =  ( ( Id `  D ) `
 ( F `  X ) ) )
2623, 25eqeq12d 2457 . . 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 255 . 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 14698 . 2  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( N ( <. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )
) )
30 eqid 2443 . . 3  |-  ( Base `  D )  =  (
Base `  D )
31 fthsect.t . . 3  |-  T  =  (Sect `  D )
3213simprd 463 . . 3  |-  ( ph  ->  D  e.  Cat )
331, 30, 9funcf1 14781 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
3433, 5ffvelrnd 5849 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
3533, 15ffvelrnd 5849 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
361, 2, 3, 9, 5, 15funcf2 14783 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) ( Hom  `  D
) ( F `  Y ) ) )
3736, 16ffvelrnd 5849 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) ( Hom  `  D
) ( F `  Y ) ) )
381, 2, 3, 9, 15, 5funcf2 14783 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y H X ) --> ( ( F `  Y
) ( Hom  `  D
) ( F `  X ) ) )
3938, 17ffvelrnd 5849 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) ( Hom  `  D
) ( F `  X ) ) )
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 14698 . 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 285 1  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( ( X G Y ) `  M
) ( ( F `
 X ) T ( F `  Y
) ) ( ( Y G X ) `
 N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   <.cop 3888   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   Hom chom 14254  compcco 14255   Catccat 14607   Idccid 14608  Sectcsect 14688    Func cfunc 14769   Faith cfth 14818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-map 7221  df-ixp 7269  df-cat 14611  df-cid 14612  df-sect 14691  df-func 14773  df-fth 14820
This theorem is referenced by:  fthinv  14841
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