Step |
Hyp |
Ref |
Expression |
1 |
|
invisoinv.b |
|- B = ( Base ` C ) |
2 |
|
invisoinv.i |
|- I = ( Iso ` C ) |
3 |
|
invisoinv.n |
|- N = ( Inv ` C ) |
4 |
|
invisoinv.c |
|- ( ph -> C e. Cat ) |
5 |
|
invisoinv.x |
|- ( ph -> X e. B ) |
6 |
|
invisoinv.y |
|- ( ph -> Y e. B ) |
7 |
|
invisoinv.f |
|- ( ph -> F e. ( X I Y ) ) |
8 |
|
invcoisoid.1 |
|- .1. = ( Id ` C ) |
9 |
|
invcoisoid.o |
|- .o. = ( <. X , Y >. ( comp ` C ) X ) |
10 |
1 2 3 4 5 6 7
|
invisoinvr |
|- ( ph -> F ( X N Y ) ( ( X N Y ) ` F ) ) |
11 |
|
eqid |
|- ( Sect ` C ) = ( Sect ` C ) |
12 |
1 3 4 5 6 11
|
isinv |
|- ( ph -> ( F ( X N Y ) ( ( X N Y ) ` F ) <-> ( F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) /\ ( ( X N Y ) ` F ) ( Y ( Sect ` C ) X ) F ) ) ) |
13 |
|
simpl |
|- ( ( F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) /\ ( ( X N Y ) ` F ) ( Y ( Sect ` C ) X ) F ) -> F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) ) |
14 |
12 13
|
syl6bi |
|- ( ph -> ( F ( X N Y ) ( ( X N Y ) ` F ) -> F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) ) ) |
15 |
10 14
|
mpd |
|- ( ph -> F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) ) |
16 |
|
eqid |
|- ( Hom ` C ) = ( Hom ` C ) |
17 |
|
eqid |
|- ( comp ` C ) = ( comp ` C ) |
18 |
1 16 2 4 5 6
|
isohom |
|- ( ph -> ( X I Y ) C_ ( X ( Hom ` C ) Y ) ) |
19 |
18 7
|
sseldd |
|- ( ph -> F e. ( X ( Hom ` C ) Y ) ) |
20 |
1 16 2 4 6 5
|
isohom |
|- ( ph -> ( Y I X ) C_ ( Y ( Hom ` C ) X ) ) |
21 |
1 3 4 5 6 2
|
invf |
|- ( ph -> ( X N Y ) : ( X I Y ) --> ( Y I X ) ) |
22 |
21 7
|
ffvelrnd |
|- ( ph -> ( ( X N Y ) ` F ) e. ( Y I X ) ) |
23 |
20 22
|
sseldd |
|- ( ph -> ( ( X N Y ) ` F ) e. ( Y ( Hom ` C ) X ) ) |
24 |
1 16 17 8 11 4 5 6 19 23
|
issect2 |
|- ( ph -> ( F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) <-> ( ( ( X N Y ) ` F ) ( <. X , Y >. ( comp ` C ) X ) F ) = ( .1. ` X ) ) ) |
25 |
9
|
a1i |
|- ( ph -> .o. = ( <. X , Y >. ( comp ` C ) X ) ) |
26 |
25
|
eqcomd |
|- ( ph -> ( <. X , Y >. ( comp ` C ) X ) = .o. ) |
27 |
26
|
oveqd |
|- ( ph -> ( ( ( X N Y ) ` F ) ( <. X , Y >. ( comp ` C ) X ) F ) = ( ( ( X N Y ) ` F ) .o. F ) ) |
28 |
27
|
eqeq1d |
|- ( ph -> ( ( ( ( X N Y ) ` F ) ( <. X , Y >. ( comp ` C ) X ) F ) = ( .1. ` X ) <-> ( ( ( X N Y ) ` F ) .o. F ) = ( .1. ` X ) ) ) |
29 |
24 28
|
bitrd |
|- ( ph -> ( F ( X ( Sect ` C ) Y ) ( ( X N Y ) ` F ) <-> ( ( ( X N Y ) ` F ) .o. F ) = ( .1. ` X ) ) ) |
30 |
15 29
|
mpbid |
|- ( ph -> ( ( ( X N Y ) ` F ) .o. F ) = ( .1. ` X ) ) |