Step |
Hyp |
Ref |
Expression |
1 |
|
pcopt.1 |
|- P = ( ( 0 [,] 1 ) X. { Y } ) |
2 |
|
iitopon |
|- II e. ( TopOn ` ( 0 [,] 1 ) ) |
3 |
|
cnconst2 |
|- ( ( II e. ( TopOn ` ( 0 [,] 1 ) ) /\ J e. ( TopOn ` X ) /\ Y e. X ) -> ( ( 0 [,] 1 ) X. { Y } ) e. ( II Cn J ) ) |
4 |
2 3
|
mp3an1 |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( ( 0 [,] 1 ) X. { Y } ) e. ( II Cn J ) ) |
5 |
1 4
|
eqeltrid |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> P e. ( II Cn J ) ) |
6 |
1
|
fveq1i |
|- ( P ` 0 ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` 0 ) |
7 |
|
simpr |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> Y e. X ) |
8 |
|
0elunit |
|- 0 e. ( 0 [,] 1 ) |
9 |
|
fvconst2g |
|- ( ( Y e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` 0 ) = Y ) |
10 |
7 8 9
|
sylancl |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` 0 ) = Y ) |
11 |
6 10
|
eqtrid |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( P ` 0 ) = Y ) |
12 |
1
|
fveq1i |
|- ( P ` 1 ) = ( ( ( 0 [,] 1 ) X. { Y } ) ` 1 ) |
13 |
|
1elunit |
|- 1 e. ( 0 [,] 1 ) |
14 |
|
fvconst2g |
|- ( ( Y e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` 1 ) = Y ) |
15 |
7 13 14
|
sylancl |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( ( ( 0 [,] 1 ) X. { Y } ) ` 1 ) = Y ) |
16 |
12 15
|
eqtrid |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( P ` 1 ) = Y ) |
17 |
5 11 16
|
3jca |
|- ( ( J e. ( TopOn ` X ) /\ Y e. X ) -> ( P e. ( II Cn J ) /\ ( P ` 0 ) = Y /\ ( P ` 1 ) = Y ) ) |