Step |
Hyp |
Ref |
Expression |
1 |
|
issect.b |
|- B = ( Base ` C ) |
2 |
|
issect.h |
|- H = ( Hom ` C ) |
3 |
|
issect.o |
|- .x. = ( comp ` C ) |
4 |
|
issect.i |
|- .1. = ( Id ` C ) |
5 |
|
issect.s |
|- S = ( Sect ` C ) |
6 |
|
issect.c |
|- ( ph -> C e. Cat ) |
7 |
|
issect.x |
|- ( ph -> X e. B ) |
8 |
|
issect.y |
|- ( ph -> Y e. B ) |
9 |
1 2 3 4 5 6 7 8
|
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 ) ) } ) |
10 |
9
|
breqd |
|- ( ph -> ( F ( X S Y ) G <-> F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G ) ) |
11 |
|
oveq12 |
|- ( ( g = G /\ f = F ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) ) |
12 |
11
|
ancoms |
|- ( ( f = F /\ g = G ) -> ( g ( <. X , Y >. .x. X ) f ) = ( G ( <. X , Y >. .x. X ) F ) ) |
13 |
12
|
eqeq1d |
|- ( ( f = F /\ g = G ) -> ( ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) <-> ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) |
14 |
|
eqid |
|- { <. 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 ) ) } |
15 |
13 14
|
brab2a |
|- ( F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G <-> ( ( F e. ( X H Y ) /\ G e. ( Y H X ) ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) |
16 |
|
df-3an |
|- ( ( 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 ) ) ) |
17 |
15 16
|
bitr4i |
|- ( F { <. f , g >. | ( ( f e. ( X H Y ) /\ g e. ( Y H X ) ) /\ ( g ( <. X , Y >. .x. X ) f ) = ( .1. ` X ) ) } G <-> ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) |
18 |
10 17
|
bitrdi |
|- ( ph -> ( F ( X S Y ) G <-> ( F e. ( X H Y ) /\ G e. ( Y H X ) /\ ( G ( <. X , Y >. .x. X ) F ) = ( .1. ` X ) ) ) ) |