Theorem sectfval 16411

Description: Value of the section relation. (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
sectfval	|- ( ph -> ( X S Y ) = { <. 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, .1.   C, f, g   ph, f, g   f, H, g   .x. , f, g   f, X, g   f, Y, g

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

Proof of Theorem sectfval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issect.b	. . 3 |- B = ( Base ` C )
2		issect.h	. . 3 |- H = ( Hom ` C )
3		issect.o	. . 3 |- .x. = (comp ` C )
4		issect.i	. . 3 |- .1. = ( Id ` C )
5		issect.s	. . 3 |- S = (Sect ` C )
6		issect.c	. . 3 |- ( ph -> C e. Cat )
7		issect.x	. . 3 |- ( ph -> X e. B )
8	1, 2, 3, 4, 5, 6, 7, 7	sectffval 16410	. 2 |- ( 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 ) ) } ) )
9		simprl 794	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> x = X )
10		simprr 796	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> y = Y )
11	9, 10	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x H y ) = ( X H Y ) )
12	11	eleq2d 2687	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( f e. ( x H y ) <-> f e. ( X H Y ) ) )
13	10, 9	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( y H x ) = ( Y H X ) )
14	13	eleq2d 2687	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( g e. ( y H x ) <-> g e. ( Y H X ) ) )
15	12, 14	anbi12d 747	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( f e. ( x H y ) /\ g e. ( y H x ) ) <-> ( f e. ( X H Y ) /\ g e. ( Y H X ) ) ) )
16	9, 10	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> <. x , y >. = <. X , Y >. )
17	16, 9	oveq12d 6668	. . . . . 6 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( <. x , y >. .x. x ) = ( <. X , Y >. .x. X ) )
18	17	oveqd 6667	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( g ( <. x , y >. .x. x ) f ) = ( g ( <. X , Y >. .x. X ) f ) )
19	9	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( .1. ` x ) = ( .1. ` X ) )
20	18, 19	eqeq12d 2637	. . . 4 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) <-> ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) )
21	15, 20	anbi12d 747	. . 3 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) <-> ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) ) )
22	21	opabbidv 4716	. 2 |- ( ( ph /\ ( x = X /\ y = Y ) ) -> { <. f , g >. | ( ( f e. ( x H y ) /\ g e. ( y H x ) ) /\ ( g ( <. x , y >. .x. x ) f ) = ( .1. ` x ) ) } = { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } )
23		issect.y	. 2 |- ( ph -> Y e. B )
24		ovex 6678	. . . . 5 |- ( X H Y ) e. _V
25		ovex 6678	. . . . 5 |- ( Y H X ) e. _V
26	24, 25	xpex 6962	. . . 4 |- ( ( X H Y ) X. ( Y H X ) ) e. _V
27		opabssxp 5193	. . . 4 |- { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } C_ ( ( X H Y ) X. ( Y H X ) )
28	26, 27	ssexi 4803	. . 3 |- { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } e. _V
29	28	a1i 11	. 2 |- ( ph -> { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } e. _V )
30	8, 22, 7, 23, 29	ovmpt2d 6788	1 |- ( ph -> ( X S Y ) = { <. 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   <.cop 4183   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   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:  sectss  16412  issect  16413  dfiso2  16432