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Theorem funcsect 14074
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 2438 . . . . . . 7  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2438 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5 eqid 2438 . . . . . . 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 4216 . . . . . . . . . 10  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
97, 8sylib 190 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
10 funcrcl 14065 . . . . . . . . 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 447 . . . . . . 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 13984 . . . . . 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 203 . . . . 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 972 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  D ) X ) M )  =  ( ( Id `  D
) `  X )
)
1817fveq2d 5735 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  D ) `
 X ) ) )
19 eqid 2438 . . . 4  |-  (comp `  E )  =  (comp `  E )
2016simp1d 970 . . . 4  |-  ( ph  ->  M  e.  ( X (  Hom  `  D
) Y ) )
2116simp2d 971 . . . 4  |-  ( ph  ->  N  e.  ( Y (  Hom  `  D
) X ) )
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 14073 . . 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 2438 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
242, 5, 23, 7, 13funcid 14072 . . 3  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  D
) `  X )
)  =  ( ( Id `  E ) `
 ( F `  X ) ) )
2518, 22, 243eqtr3d 2478 . 2  |-  ( ph  ->  ( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) )
26 eqid 2438 . . 3  |-  ( Base `  E )  =  (
Base `  E )
27 eqid 2438 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
28 funcsect.t . . 3  |-  T  =  (Sect `  E )
2911simprd 451 . . 3  |-  ( ph  ->  E  e.  Cat )
302, 26, 7funcf1 14068 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
3130, 13ffvelrnd 5874 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
3230, 14ffvelrnd 5874 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
332, 3, 27, 7, 13, 14funcf2 14070 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X (  Hom  `  D
) Y ) --> ( ( F `  X
) (  Hom  `  E
) ( F `  Y ) ) )
3433, 20ffvelrnd 5874 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  E
) ( F `  Y ) ) )
352, 3, 27, 7, 14, 13funcf2 14070 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y (  Hom  `  D
) X ) --> ( ( F `  Y
) (  Hom  `  E
) ( F `  X ) ) )
3635, 21ffvelrnd 5874 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  E
) ( F `  X ) ) )
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 13985 . 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 225 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   <.cop 3819   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894   Idccid 13895  Sectcsect 13975    Func cfunc 14056
This theorem is referenced by:  funcinv  14075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-ixp 7067  df-sect 13978  df-func 14060
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