Theorem funcsect 16532

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

Step	Hyp	Ref	Expression
1		funcsect.m	. . . . . 6 |- ( ph -> M ( X S Y ) N )
2		funcsect.b	. . . . . . 7 |- B = ( Base ` D )
3		eqid 2622	. . . . . . 7 |- ( Hom ` D ) = ( Hom ` D )
4		eqid 2622	. . . . . . 7 |- (comp ` D ) = (comp ` D )
5		eqid 2622	. . . . . . 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 4654	. . . . . . . . . 10 |- ( F ( D Func E ) G <-> <. F , G >. e. ( D Func E ) )
9	7, 8	sylib 208	. . . . . . . . 9 |- ( ph -> <. F , G >. e. ( D Func E ) )
10		funcrcl 16523	. . . . . . . . 9 |- ( <. F , G >. e. ( D Func E ) -> ( D e. Cat /\ E e. Cat ) )
11	9, 10	syl 17	. . . . . . . 8 |- ( ph -> ( D e. Cat /\ E e. Cat ) )
12	11	simpld 475	. . . . . . 7 |- ( ph -> D e. Cat )
13		funcsect.x	. . . . . . 7 |- ( ph -> X e. B )
14		funcsect.y	. . . . . . 7 |- ( ph -> Y e. B )
15	2, 3, 4, 5, 6, 12, 13, 14	issect 16413	. . . . . 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 ) ) ) )
16	1, 15	mpbid 222	. . . . 5 |- ( ph -> ( M e. ( X ( Hom ` D ) Y ) /\ N e. ( Y ( Hom ` D ) X ) /\ ( N ( <. X , Y >. (comp ` D ) X ) M ) = ( ( Id ` D ) ` X ) ) )
17	16	simp3d 1075	. . . 4 |- ( ph -> ( N ( <. X , Y >. (comp ` D ) X ) M ) = ( ( Id ` D ) ` X ) )
18	17	fveq2d 6195	. . 3 |- ( ph -> ( ( X G X ) ` ( N ( <. X , Y >. (comp ` D ) X ) M ) ) = ( ( X G X ) ` ( ( Id ` D ) ` X ) ) )
19		eqid 2622	. . . 4 |- (comp ` E ) = (comp ` E )
20	16	simp1d 1073	. . . 4 |- ( ph -> M e. ( X ( Hom ` D ) Y ) )
21	16	simp2d 1074	. . . 4 |- ( ph -> N e. ( Y ( Hom ` D ) X ) )
22	2, 3, 4, 19, 7, 13, 14, 13, 20, 21	funcco 16531	. . 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 2622	. . . 4 |- ( Id ` E ) = ( Id ` E )
24	2, 5, 23, 7, 13	funcid 16530	. . 3 |- ( ph -> ( ( X G X ) ` ( ( Id ` D ) ` X ) ) = ( ( Id ` E ) ` ( F ` X ) ) )
25	18, 22, 24	3eqtr3d 2664	. 2 |- ( ph -> ( ( ( Y G X ) ` N ) ( <. ( F ` X ) , ( F ` Y ) >. (comp ` E ) ( F ` X ) ) ( ( X G Y ) ` M ) ) = ( ( Id ` E ) ` ( F ` X ) ) )
26		eqid 2622	. . 3 |- ( Base ` E ) = ( Base ` E )
27		eqid 2622	. . 3 |- ( Hom ` E ) = ( Hom ` E )
28		funcsect.t	. . 3 |- T = (Sect ` E )
29	11	simprd 479	. . 3 |- ( ph -> E e. Cat )
30	2, 26, 7	funcf1 16526	. . . 4 |- ( ph -> F : B --> ( Base ` E ) )
31	30, 13	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` X ) e. ( Base ` E ) )
32	30, 14	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` Y ) e. ( Base ` E ) )
33	2, 3, 27, 7, 13, 14	funcf2 16528	. . . 4 |- ( ph -> ( X G Y ) : ( X ( Hom ` D ) Y ) --> ( ( F ` X ) ( Hom ` E ) ( F ` Y ) ) )
34	33, 20	ffvelrnd 6360	. . 3 |- ( ph -> ( ( X G Y ) ` M ) e. ( ( F ` X ) ( Hom ` E ) ( F ` Y ) ) )
35	2, 3, 27, 7, 14, 13	funcf2 16528	. . . 4 |- ( ph -> ( Y G X ) : ( Y ( Hom ` D ) X ) --> ( ( F ` Y ) ( Hom ` E ) ( F ` X ) ) )
36	35, 21	ffvelrnd 6360	. . 3 |- ( ph -> ( ( Y G X ) ` N ) e. ( ( F ` Y ) ( Hom ` E ) ( F ` X ) ) )
37	26, 27, 19, 23, 28, 29, 31, 32, 34, 36	issect2 16414	. 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 ) ) ) )
38	25, 37	mpbird 247	1 |- ( ph -> ( ( X G Y ) ` M ) ( ( F ` X ) T ( F ` Y ) ) ( ( Y G X ) ` N ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404   Func cfunc 16514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ixp 7909  df-sect 16407  df-func 16518

This theorem is referenced by:  funcinv  16533