Step |
Hyp |
Ref |
Expression |
1 |
|
cnmpt1ds.d |
|- D = ( dist ` G ) |
2 |
|
cnmpt1ds.j |
|- J = ( TopOpen ` G ) |
3 |
|
cnmpt1ds.r |
|- R = ( topGen ` ran (,) ) |
4 |
|
cnmpt1ds.g |
|- ( ph -> G e. MetSp ) |
5 |
|
cnmpt1ds.k |
|- ( ph -> K e. ( TopOn ` X ) ) |
6 |
|
cnmpt1ds.a |
|- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) ) |
7 |
|
cnmpt1ds.b |
|- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) ) |
8 |
|
mstps |
|- ( G e. MetSp -> G e. TopSp ) |
9 |
4 8
|
syl |
|- ( ph -> G e. TopSp ) |
10 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
11 |
10 2
|
istps |
|- ( G e. TopSp <-> J e. ( TopOn ` ( Base ` G ) ) ) |
12 |
9 11
|
sylib |
|- ( ph -> J e. ( TopOn ` ( Base ` G ) ) ) |
13 |
|
cnf2 |
|- ( ( K e. ( TopOn ` X ) /\ J e. ( TopOn ` ( Base ` G ) ) /\ ( x e. X |-> A ) e. ( K Cn J ) ) -> ( x e. X |-> A ) : X --> ( Base ` G ) ) |
14 |
5 12 6 13
|
syl3anc |
|- ( ph -> ( x e. X |-> A ) : X --> ( Base ` G ) ) |
15 |
14
|
fvmptelrn |
|- ( ( ph /\ x e. X ) -> A e. ( Base ` G ) ) |
16 |
|
cnf2 |
|- ( ( K e. ( TopOn ` X ) /\ J e. ( TopOn ` ( Base ` G ) ) /\ ( x e. X |-> B ) e. ( K Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` G ) ) |
17 |
5 12 7 16
|
syl3anc |
|- ( ph -> ( x e. X |-> B ) : X --> ( Base ` G ) ) |
18 |
17
|
fvmptelrn |
|- ( ( ph /\ x e. X ) -> B e. ( Base ` G ) ) |
19 |
15 18
|
ovresd |
|- ( ( ph /\ x e. X ) -> ( A ( D |` ( ( Base ` G ) X. ( Base ` G ) ) ) B ) = ( A D B ) ) |
20 |
19
|
mpteq2dva |
|- ( ph -> ( x e. X |-> ( A ( D |` ( ( Base ` G ) X. ( Base ` G ) ) ) B ) ) = ( x e. X |-> ( A D B ) ) ) |
21 |
10 1 2 3
|
msdcn |
|- ( G e. MetSp -> ( D |` ( ( Base ` G ) X. ( Base ` G ) ) ) e. ( ( J tX J ) Cn R ) ) |
22 |
4 21
|
syl |
|- ( ph -> ( D |` ( ( Base ` G ) X. ( Base ` G ) ) ) e. ( ( J tX J ) Cn R ) ) |
23 |
5 6 7 22
|
cnmpt12f |
|- ( ph -> ( x e. X |-> ( A ( D |` ( ( Base ` G ) X. ( Base ` G ) ) ) B ) ) e. ( K Cn R ) ) |
24 |
20 23
|
eqeltrrd |
|- ( ph -> ( x e. X |-> ( A D B ) ) e. ( K Cn R ) ) |