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Theorem fthsect 15152
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 2467 . . . 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 2467 . . . . 5  |-  (comp `  C )  =  (comp `  C )
7 fthfunc 15134 . . . . . . . . . 10  |-  ( C Faith 
D )  C_  ( C  Func  D )
87ssbri 4489 . . . . . . . . 9  |-  ( F ( C Faith  D ) G  ->  F ( C  Func  D ) G )
94, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F ( C  Func  D ) G )
10 df-br 4448 . . . . . . . 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 15090 . . . . . . 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 14940 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  C ) X ) M )  e.  ( X H X ) )
19 eqid 2467 . . . . 5  |-  ( Id
`  C )  =  ( Id `  C
)
201, 2, 19, 14, 5catidcl 14937 . . . 4  |-  ( ph  ->  ( ( Id `  C ) `  X
)  e.  ( X H X ) )
211, 2, 3, 4, 5, 5, 18, 20fthi 15145 . . 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 2467 . . . . 5  |-  (comp `  D )  =  (comp `  D )
231, 2, 6, 22, 9, 5, 15, 5, 16, 17funcco 15098 . . . 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 2467 . . . . 5  |-  ( Id
`  D )  =  ( Id `  D
)
251, 19, 24, 9, 5funcid 15097 . . . 4  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  C
) `  X )
)  =  ( ( Id `  D ) `
 ( F `  X ) ) )
2623, 25eqeq12d 2489 . . 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 15010 . 2  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( N ( <. X ,  Y >. (comp `  C ) X ) M )  =  ( ( Id `  C
) `  X )
) )
30 eqid 2467 . . 3  |-  ( Base `  D )  =  (
Base `  D )
31 fthsect.t . . 3  |-  T  =  (Sect `  D )
3213simprd 463 . . 3  |-  ( ph  ->  D  e.  Cat )
331, 30, 9funcf1 15093 . . . 4  |-  ( ph  ->  F : B --> ( Base `  D ) )
3433, 5ffvelrnd 6022 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  D ) )
3533, 15ffvelrnd 6022 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  D ) )
361, 2, 3, 9, 5, 15funcf2 15095 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X H Y ) --> ( ( F `  X
) ( Hom  `  D
) ( F `  Y ) ) )
3736, 16ffvelrnd 6022 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) ( Hom  `  D
) ( F `  Y ) ) )
381, 2, 3, 9, 15, 5funcf2 15095 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y H X ) --> ( ( F `  Y
) ( Hom  `  D
) ( F `  X ) ) )
3938, 17ffvelrnd 6022 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) ( Hom  `  D
) ( F `  X ) ) )
4030, 3, 22, 24, 31, 32, 34, 35, 37, 39issect2 15010 . 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 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919   Idccid 14920  Sectcsect 15000    Func cfunc 15081   Faith cfth 15130
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-map 7422  df-ixp 7470  df-cat 14923  df-cid 14924  df-sect 15003  df-func 15085  df-fth 15132
This theorem is referenced by:  fthinv  15153
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