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Theorem funcsect 15095
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 2467 . . . . . . 7  |-  ( Hom  `  D )  =  ( Hom  `  D )
4 eqid 2467 . . . . . . 7  |-  (comp `  D )  =  (comp `  D )
5 eqid 2467 . . . . . . 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 4448 . . . . . . . . . 10  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
97, 8sylib 196 . . . . . . . . 9  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
10 funcrcl 15086 . . . . . . . . 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 459 . . . . . . 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 15005 . . . . . 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 210 . . . . 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 1010 . . . 4  |-  ( ph  ->  ( N ( <. X ,  Y >. (comp `  D ) X ) M )  =  ( ( Id `  D
) `  X )
)
1817fveq2d 5868 . . 3  |-  ( ph  ->  ( ( X G X ) `  ( N ( <. X ,  Y >. (comp `  D
) X ) M ) )  =  ( ( X G X ) `  ( ( Id `  D ) `
 X ) ) )
19 eqid 2467 . . . 4  |-  (comp `  E )  =  (comp `  E )
2016simp1d 1008 . . . 4  |-  ( ph  ->  M  e.  ( X ( Hom  `  D
) Y ) )
2116simp2d 1009 . . . 4  |-  ( ph  ->  N  e.  ( Y ( Hom  `  D
) X ) )
222, 3, 4, 19, 7, 13, 14, 13, 20, 21funcco 15094 . . 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 2467 . . . 4  |-  ( Id
`  E )  =  ( Id `  E
)
242, 5, 23, 7, 13funcid 15093 . . 3  |-  ( ph  ->  ( ( X G X ) `  (
( Id `  D
) `  X )
)  =  ( ( Id `  E ) `
 ( F `  X ) ) )
2518, 22, 243eqtr3d 2516 . 2  |-  ( ph  ->  ( ( ( Y G X ) `  N ) ( <.
( F `  X
) ,  ( F `
 Y ) >.
(comp `  E )
( F `  X
) ) ( ( X G Y ) `
 M ) )  =  ( ( Id
`  E ) `  ( F `  X ) ) )
26 eqid 2467 . . 3  |-  ( Base `  E )  =  (
Base `  E )
27 eqid 2467 . . 3  |-  ( Hom  `  E )  =  ( Hom  `  E )
28 funcsect.t . . 3  |-  T  =  (Sect `  E )
2911simprd 463 . . 3  |-  ( ph  ->  E  e.  Cat )
302, 26, 7funcf1 15089 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
3130, 13ffvelrnd 6020 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
3230, 14ffvelrnd 6020 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
332, 3, 27, 7, 13, 14funcf2 15091 . . . 4  |-  ( ph  ->  ( X G Y ) : ( X ( Hom  `  D
) Y ) --> ( ( F `  X
) ( Hom  `  E
) ( F `  Y ) ) )
3433, 20ffvelrnd 6020 . . 3  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) ( Hom  `  E
) ( F `  Y ) ) )
352, 3, 27, 7, 14, 13funcf2 15091 . . . 4  |-  ( ph  ->  ( Y G X ) : ( Y ( Hom  `  D
) X ) --> ( ( F `  Y
) ( Hom  `  E
) ( F `  X ) ) )
3635, 21ffvelrnd 6020 . . 3  |-  ( ph  ->  ( ( Y G X ) `  N
)  e.  ( ( F `  Y ) ( Hom  `  E
) ( F `  X ) ) )
3726, 27, 19, 23, 28, 29, 31, 32, 34, 36issect2 15006 . 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 232 1  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) T ( F `  Y
) ) ( ( Y G X ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14486   Hom chom 14562  compcco 14563   Catccat 14915   Idccid 14916  Sectcsect 14996    Func cfunc 15077
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 6574
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-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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-ixp 7467  df-sect 14999  df-func 15081
This theorem is referenced by:  funcinv  15096
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