MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcsect Unicode version

Theorem funcsect 13746
Description: The image of a section under a functor is a section. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcsect.b  |-  B  =  ( Base `  D
)
funcsect.s  |-  S  =  (Sect `  D )
funcsect.t  |-  T  =  (Sect `  E )
funcsect.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcsect.x  |-  ( ph  ->  X  e.  B )
funcsect.y  |-  ( ph  ->  Y  e.  B )
funcsect.m  |-  ( ph  ->  M ( X S Y ) N )
Assertion
Ref Expression
funcsect  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) T ( F `  Y
) ) ( ( Y G X ) `
 N ) )

Proof of Theorem funcsect
StepHypRef Expression
1 funcsect.m . . . . . 6  |-  ( ph  ->  M ( X S Y ) N )
2 funcsect.b . . . . . . 7  |-  B  =  ( Base `  D
)
3 eqid 2283 . . . . . . 7  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2283 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5 eqid 2283 . . . . . . 7  |-  ( Id
`  D )  =  ( Id `  D
)
6 funcsect.s . . . . . . 7  |-  S  =  (Sect `  D )
7 funcsect.f . . . . . . . . . 10  |-  ( ph  ->  F ( D  Func  E ) G )
8 df-br 4024 . . . . . . . . . 10  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
97, 8sylib 188 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
10 funcrcl 13737 . . . . . . . . 9  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
119, 10syl 15 . . . . . . . 8  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1211simpld 445 . . . . . . 7  |-  ( ph  ->  D  e.  Cat )
13 funcsect.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
14 funcsect.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
152, 3, 4, 5, 6, 12, 13, 14issect 13656 . . . . . 6  |-  ( ph  ->  ( M ( X S Y ) N  <-> 
( M  e.  ( X (  Hom  `  D
) Y )  /\  N  e.  ( Y
(  Hom  `  D ) X )  /\  ( N ( <. X ,  Y >. (comp `  D
) X ) M )  =  ( ( Id `  D ) `
 X ) ) ) )
161, 15mpbid 201 . . . . 5  |-  ( ph  ->  ( M  e.  ( X (  Hom  `  D
) Y )  /\  N  e.  ( Y
(  Hom  `  D ) X )  /\  ( N ( <. X ,  Y >. (comp `  D
) X ) M )  =  ( ( Id `  D ) `
 X ) ) )
1716simp3d 969 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  D ) X ) M )  =  ( ( Id `  D
) `  X )
)
1817fveq2d 5529 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  D ) `
 X ) ) )
19 eqid 2283 . . . 4  |-  (comp `  E )  =  (comp `  E )
2016simp1d 967 . . . 4  |-  ( ph  ->  M  e.  ( X (  Hom  `  D
) Y ) )
2116simp2d 968 . . . 4  |-  ( ph  ->  N  e.  ( Y (  Hom  `  D
) X ) )
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 13745 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( ( Y G X ) `  N
) ( <. ( F `  X ) ,  ( F `  Y ) >. (comp `  E ) ( F `
 X ) ) ( ( X G Y ) `  M
) ) )
23 eqid 2283 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
242, 5, 23, 7, 13funcid 13744 . . 3  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  D
) `  X )
)  =  ( ( Id `  E ) `
 ( F `  X ) ) )
2518, 22, 243eqtr3d 2323 . 2  |-  ( ph  ->  ( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) )
26 eqid 2283 . . 3  |-  ( Base `  E )  =  (
Base `  E )
27 eqid 2283 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
28 funcsect.t . . 3  |-  T  =  (Sect `  E )
2911simprd 449 . . 3  |-  ( ph  ->  E  e.  Cat )
302, 26, 7funcf1 13740 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
3130, 13ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
3230, 14ffvelrnd 5666 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
332, 3, 27, 7, 13, 14funcf2 13742 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X (  Hom  `  D
) Y ) --> ( ( F `  X
) (  Hom  `  E
) ( F `  Y ) ) )
3433, 20ffvelrnd 5666 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  E
) ( F `  Y ) ) )
352, 3, 27, 7, 14, 13funcf2 13742 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y (  Hom  `  D
) X ) --> ( ( F `  Y
) (  Hom  `  E
) ( F `  X ) ) )
3635, 21ffvelrnd 5666 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  E
) ( F `  X ) ) )
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 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 `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) ) )
3825, 37mpbird 223 1  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) T ( F `  Y
) ) ( ( Y G X ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = 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
This theorem is referenced by:  funcinv  13747
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-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-map 6774  df-ixp 6818  df-sect 13650  df-func 13732
  Copyright terms: Public domain W3C validator