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Theorem funcsect 13956
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 2366 . . . . . . 7  |-  (  Hom  `  D )  =  (  Hom  `  D )
4 eqid 2366 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5 eqid 2366 . . . . . . 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 4126 . . . . . . . . . 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 13947 . . . . . . . . 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 13866 . . . . . 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 970 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  D ) X ) M )  =  ( ( Id `  D
) `  X )
)
1817fveq2d 5636 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  D ) `
 X ) ) )
19 eqid 2366 . . . 4  |-  (comp `  E )  =  (comp `  E )
2016simp1d 968 . . . 4  |-  ( ph  ->  M  e.  ( X (  Hom  `  D
) Y ) )
2116simp2d 969 . . . 4  |-  ( ph  ->  N  e.  ( Y (  Hom  `  D
) X ) )
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 13955 . . 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 2366 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
242, 5, 23, 7, 13funcid 13954 . . 3  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  D
) `  X )
)  =  ( ( Id `  E ) `
 ( F `  X ) ) )
2518, 22, 243eqtr3d 2406 . 2  |-  ( ph  ->  ( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) )
26 eqid 2366 . . 3  |-  ( Base `  E )  =  (
Base `  E )
27 eqid 2366 . . 3  |-  (  Hom  `  E )  =  (  Hom  `  E )
28 funcsect.t . . 3  |-  T  =  (Sect `  E )
2911simprd 449 . . 3  |-  ( ph  ->  E  e.  Cat )
302, 26, 7funcf1 13950 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
3130, 13ffvelrnd 5773 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
3230, 14ffvelrnd 5773 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
332, 3, 27, 7, 13, 14funcf2 13952 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X (  Hom  `  D
) Y ) --> ( ( F `  X
) (  Hom  `  E
) ( F `  Y ) ) )
3433, 20ffvelrnd 5773 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) (  Hom  `  E
) ( F `  Y ) ) )
352, 3, 27, 7, 14, 13funcf2 13952 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y (  Hom  `  D
) X ) --> ( ( F `  Y
) (  Hom  `  E
) ( F `  X ) ) )
3635, 21ffvelrnd 5773 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) (  Hom  `  E
) ( F `  X ) ) )
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 13867 . 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 935    = wceq 1647    e. wcel 1715   <.cop 3732   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   Basecbs 13356    Hom chom 13427  compcco 13428   Catccat 13776   Idccid 13777  Sectcsect 13857    Func cfunc 13938
This theorem is referenced by:  funcinv  13957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-map 6917  df-ixp 6961  df-sect 13860  df-func 13942
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