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Theorem funcsect 14024
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 2404 . . . . . . 7  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2404 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5 eqid 2404 . . . . . . 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 4173 . . . . . . . . . 10  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
97, 8sylib 189 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
10 funcrcl 14015 . . . . . . . . 9  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
119, 10syl 16 . . . . . . . 8  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1211simpld 446 . . . . . . 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 13934 . . . . . 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 202 . . . . 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 971 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  D ) X ) M )  =  ( ( Id `  D
) `  X )
)
1817fveq2d 5691 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  D ) `
 X ) ) )
19 eqid 2404 . . . 4  |-  (comp `  E )  =  (comp `  E )
2016simp1d 969 . . . 4  |-  ( ph  ->  M  e.  ( X (  Hom  `  D
) Y ) )
2116simp2d 970 . . . 4  |-  ( ph  ->  N  e.  ( Y (  Hom  `  D
) X ) )
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 14023 . . 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 2404 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
242, 5, 23, 7, 13funcid 14022 . . 3  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  D
) `  X )
)  =  ( ( Id `  E ) `
 ( F `  X ) ) )
2518, 22, 243eqtr3d 2444 . 2  |-  ( ph  ->  ( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) )
26 eqid 2404 . . 3  |-  ( Base `  E )  =  (
Base `  E )
27 eqid 2404 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
28 funcsect.t . . 3  |-  T  =  (Sect `  E )
2911simprd 450 . . 3  |-  ( ph  ->  E  e.  Cat )
302, 26, 7funcf1 14018 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
3130, 13ffvelrnd 5830 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
3230, 14ffvelrnd 5830 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
332, 3, 27, 7, 13, 14funcf2 14020 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X (  Hom  `  D
) Y ) --> ( ( F `  X
) (  Hom  `  E
) ( F `  Y ) ) )
3433, 20ffvelrnd 5830 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  E
) ( F `  Y ) ) )
352, 3, 27, 7, 14, 13funcf2 14020 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y (  Hom  `  D
) X ) --> ( ( F `  Y
) (  Hom  `  E
) ( F `  X ) ) )
3635, 21ffvelrnd 5830 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  E
) ( F `  X ) ) )
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 13935 . 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 224 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845  Sectcsect 13925    Func cfunc 14006
This theorem is referenced by:  funcinv  14025
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-map 6979  df-ixp 7023  df-sect 13928  df-func 14010
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