| Step |
Hyp |
Ref |
Expression |
| 1 |
|
oppcsect.b |
|- B = ( Base ` C ) |
| 2 |
|
oppcsect.o |
|- O = ( oppCat ` C ) |
| 3 |
|
oppcsect.c |
|- ( ph -> C e. Cat ) |
| 4 |
|
oppcsect.x |
|- ( ph -> X e. B ) |
| 5 |
|
oppcsect.y |
|- ( ph -> Y e. B ) |
| 6 |
|
oppcsect.s |
|- S = ( Sect ` C ) |
| 7 |
|
oppcsect.t |
|- T = ( Sect ` O ) |
| 8 |
2 1
|
oppcbas |
|- B = ( Base ` O ) |
| 9 |
|
eqid |
|- ( Hom ` O ) = ( Hom ` O ) |
| 10 |
|
eqid |
|- ( comp ` O ) = ( comp ` O ) |
| 11 |
|
eqid |
|- ( Id ` O ) = ( Id ` O ) |
| 12 |
2
|
oppccat |
|- ( C e. Cat -> O e. Cat ) |
| 13 |
3 12
|
syl |
|- ( ph -> O e. Cat ) |
| 14 |
8 9 10 11 7 13 4 5
|
sectss |
|- ( ph -> ( X T Y ) C_ ( ( X ( Hom ` O ) Y ) X. ( Y ( Hom ` O ) X ) ) ) |
| 15 |
|
relxp |
|- Rel ( ( X ( Hom ` O ) Y ) X. ( Y ( Hom ` O ) X ) ) |
| 16 |
|
relss |
|- ( ( X T Y ) C_ ( ( X ( Hom ` O ) Y ) X. ( Y ( Hom ` O ) X ) ) -> ( Rel ( ( X ( Hom ` O ) Y ) X. ( Y ( Hom ` O ) X ) ) -> Rel ( X T Y ) ) ) |
| 17 |
14 15 16
|
mpisyl |
|- ( ph -> Rel ( X T Y ) ) |
| 18 |
|
relcnv |
|- Rel `' ( X S Y ) |
| 19 |
18
|
a1i |
|- ( ph -> Rel `' ( X S Y ) ) |
| 20 |
1 2 3 4 5 6 7
|
oppcsect |
|- ( ph -> ( f ( X T Y ) g <-> g ( X S Y ) f ) ) |
| 21 |
|
vex |
|- f e. _V |
| 22 |
|
vex |
|- g e. _V |
| 23 |
21 22
|
brcnv |
|- ( f `' ( X S Y ) g <-> g ( X S Y ) f ) |
| 24 |
20 23
|
bitr4di |
|- ( ph -> ( f ( X T Y ) g <-> f `' ( X S Y ) g ) ) |
| 25 |
17 19 24
|
eqbrrdv |
|- ( ph -> ( X T Y ) = `' ( X S Y ) ) |