Theorem sectffval 16410

Description: Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
issect.b	|- B = ( Base ` C )
issect.h	|- H = ( Hom ` C )
issect.o	|- .x. = (comp ` C )
issect.i	|- .1. = ( Id ` C )
issect.s	|- S = (Sect ` C )
issect.c	|- ( ph -> C e. Cat )
issect.x	|- ( ph -> X e. B )
issect.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
sectffval	|- ( ph -> S = ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) )

Distinct variable groups:   f, g, x, y, .1.   x, B, y   C, f, g, x, y   ph, f, g, x, y   f, H, g, x, y   .x. , f, g, x, y   f, X, g, x, y   f, Y, g, x, y

Allowed substitution hints:   B( f, g)   S( x, y, f, g)

Proof of Theorem sectffval

Dummy variables c h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.s	. 2 |- S = (Sect ` C )
2		issect.c	. . 3 |- ( ph -> C e. Cat )
3		fveq2 6191	. . . . . 6 |- ( c = C -> ( Base ` c ) = ( Base ` C ) )
4		issect.b	. . . . . 6 |- B = ( Base ` C )
5	3, 4	syl6eqr 2674	. . . . 5 |- ( c = C -> ( Base ` c ) = B )
6		fvexd 6203	. . . . . . 7 |- ( c = C -> ( Hom ` c ) e. _V )
7		fveq2 6191	. . . . . . . 8 |- ( c = C -> ( Hom ` c ) = ( Hom ` C ) )
8		issect.h	. . . . . . . 8 |- H = ( Hom ` C )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( c = C -> ( Hom ` c ) = H )
10		simpr 477	. . . . . . . . . . 11 |- ( ( c = C /\ h = H ) -> h = H )
11	10	oveqd 6667	. . . . . . . . . 10 |- ( ( c = C /\ h = H ) -> ( x h y ) = ( x H y ) )
12	11	eleq2d 2687	. . . . . . . . 9 |- ( ( c = C /\ h = H ) -> ( f e. ( x h y ) <-> f e. ( x H y ) ) )
13	10	oveqd 6667	. . . . . . . . . 10 |- ( ( c = C /\ h = H ) -> ( y h x ) = ( y H x ) )
14	13	eleq2d 2687	. . . . . . . . 9 |- ( ( c = C /\ h = H ) -> ( g e. ( y h x ) <-> g e. ( y H x ) ) )
15	12, 14	anbi12d 747	. . . . . . . 8 |- ( ( c = C /\ h = H ) -> ( ( f e. ( x h y ) /\ g e. ( y h x ) ) <-> ( f e. ( x H y ) /\ g e. ( y H x ) ) ) )
16		simpl 473	. . . . . . . . . . . . 13 |- ( ( c = C /\ h = H ) -> c = C )
17	16	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( c = C /\ h = H ) -> (comp ` c ) = (comp ` C ) )
18		issect.o	. . . . . . . . . . . 12 |- .x. = (comp ` C )
19	17, 18	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( c = C /\ h = H ) -> (comp ` c ) = .x. )
20	19	oveqd 6667	. . . . . . . . . 10 |- ( ( c = C /\ h = H ) -> ( <. x , y >. (comp ` c ) x ) = ( <. x , y >. .x. x ) )
21	20	oveqd 6667	. . . . . . . . 9 |- ( ( c = C /\ h = H ) -> ( g ( <. x , y >. (comp ` c ) x ) f ) = ( g ( <. x , y >. .x. x ) f ) )
22	16	fveq2d 6195	. . . . . . . . . . 11 |- ( ( c = C /\ h = H ) -> ( Id ` c ) = ( Id ` C ) )
23		issect.i	. . . . . . . . . . 11 |- .1. = ( Id ` C )
24	22, 23	syl6eqr 2674	. . . . . . . . . 10 |- ( ( c = C /\ h = H ) -> ( Id ` c ) = .1. )
25	24	fveq1d 6193	. . . . . . . . 9 |- ( ( c = C /\ h = H ) -> ( ( Id ` c ) ` x ) = ( .1. ` x ) )
26	21, 25	eqeq12d 2637	. . . . . . . 8 |- ( ( c = C /\ h = H ) -> ( ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) <-> ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) )
27	15, 26	anbi12d 747	. . . . . . 7 |- ( ( c = C /\ h = H ) -> ( ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) <-> ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) ) )
28	6, 9, 27	sbcied2 3473	. . . . . 6 |- ( c = C -> ( [. ( Hom ` c ) / h ]. ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) <-> ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) ) )
29	28	opabbidv 4716	. . . . 5 |- ( c = C -> { <. f , g >. | [. ( Hom ` c ) / h ]. ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) } = { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } )
30	5, 5, 29	mpt2eq123dv 6717	. . . 4 |- ( c = C -> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> { <. f , g >. | [. ( Hom ` c ) / h ]. ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) } ) = ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) )
31		df-sect 16407	. . . 4 |- Sect = ( c e. Cat |-> ( x e. ( Base ` c ) , y e. ( Base ` c ) |-> { <. f , g >. | [. ( Hom ` c ) / h ]. ( ( f e. ( x h y ) /\ g e. ( y h x ) ) /\ ( g ( <. x , y >. (comp ` c ) x ) f ) = ( ( Id ` c ) ` x ) ) } ) )
32		fvex 6201	. . . . . 6 |- ( Base ` C ) e. _V
33	4, 32	eqeltri 2697	. . . . 5 |- B e. _V
34	33, 33	mpt2ex 7247	. . . 4 |- ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) e. _V
35	30, 31, 34	fvmpt 6282	. . 3 |- ( C e. Cat -> (Sect ` C ) = ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) )
36	2, 35	syl 17	. 2 |- ( ph -> (Sect ` C ) = ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) )
37	1, 36	syl5eq 2668	1 |- ( ph -> S = ( x e. B , y e. B |-> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435   <.cop 4183   {copab 4712   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326  Sectcsect 16404

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-sect 16407

This theorem is referenced by:  sectfval  16411