| Step |
Hyp |
Ref |
Expression |
| 1 |
|
xpccat.t |
|- T = ( C Xc. D ) |
| 2 |
|
xpccat.c |
|- ( ph -> C e. Cat ) |
| 3 |
|
xpccat.d |
|- ( ph -> D e. Cat ) |
| 4 |
|
xpccat.x |
|- X = ( Base ` C ) |
| 5 |
|
xpccat.y |
|- Y = ( Base ` D ) |
| 6 |
|
xpccat.i |
|- I = ( Id ` C ) |
| 7 |
|
xpccat.j |
|- J = ( Id ` D ) |
| 8 |
|
xpcid.1 |
|- .1. = ( Id ` T ) |
| 9 |
|
xpcid.r |
|- ( ph -> R e. X ) |
| 10 |
|
xpcid.s |
|- ( ph -> S e. Y ) |
| 11 |
|
df-ov |
|- ( R .1. S ) = ( .1. ` <. R , S >. ) |
| 12 |
1 2 3 4 5 6 7
|
xpccatid |
|- ( ph -> ( T e. Cat /\ ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) ) |
| 13 |
12
|
simprd |
|- ( ph -> ( Id ` T ) = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) |
| 14 |
8 13
|
eqtrid |
|- ( ph -> .1. = ( x e. X , y e. Y |-> <. ( I ` x ) , ( J ` y ) >. ) ) |
| 15 |
|
simprl |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> x = R ) |
| 16 |
15
|
fveq2d |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> ( I ` x ) = ( I ` R ) ) |
| 17 |
|
simprr |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> y = S ) |
| 18 |
17
|
fveq2d |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> ( J ` y ) = ( J ` S ) ) |
| 19 |
16 18
|
opeq12d |
|- ( ( ph /\ ( x = R /\ y = S ) ) -> <. ( I ` x ) , ( J ` y ) >. = <. ( I ` R ) , ( J ` S ) >. ) |
| 20 |
|
opex |
|- <. ( I ` R ) , ( J ` S ) >. e. _V |
| 21 |
20
|
a1i |
|- ( ph -> <. ( I ` R ) , ( J ` S ) >. e. _V ) |
| 22 |
14 19 9 10 21
|
ovmpod |
|- ( ph -> ( R .1. S ) = <. ( I ` R ) , ( J ` S ) >. ) |
| 23 |
11 22
|
eqtr3id |
|- ( ph -> ( .1. ` <. R , S >. ) = <. ( I ` R ) , ( J ` S ) >. ) |