Step |
Hyp |
Ref |
Expression |
1 |
|
pmapfval.b |
|- B = ( Base ` K ) |
2 |
|
pmapfval.l |
|- .<_ = ( le ` K ) |
3 |
|
pmapfval.a |
|- A = ( Atoms ` K ) |
4 |
|
pmapfval.m |
|- M = ( pmap ` K ) |
5 |
1 2 3 4
|
pmapfval |
|- ( K e. C -> M = ( x e. B |-> { a e. A | a .<_ x } ) ) |
6 |
5
|
fveq1d |
|- ( K e. C -> ( M ` X ) = ( ( x e. B |-> { a e. A | a .<_ x } ) ` X ) ) |
7 |
|
breq2 |
|- ( x = X -> ( a .<_ x <-> a .<_ X ) ) |
8 |
7
|
rabbidv |
|- ( x = X -> { a e. A | a .<_ x } = { a e. A | a .<_ X } ) |
9 |
|
eqid |
|- ( x e. B |-> { a e. A | a .<_ x } ) = ( x e. B |-> { a e. A | a .<_ x } ) |
10 |
3
|
fvexi |
|- A e. _V |
11 |
10
|
rabex |
|- { a e. A | a .<_ X } e. _V |
12 |
8 9 11
|
fvmpt |
|- ( X e. B -> ( ( x e. B |-> { a e. A | a .<_ x } ) ` X ) = { a e. A | a .<_ X } ) |
13 |
6 12
|
sylan9eq |
|- ( ( K e. C /\ X e. B ) -> ( M ` X ) = { a e. A | a .<_ X } ) |