Step |
Hyp |
Ref |
Expression |
1 |
|
mrcfval.f |
|- F = ( mrCls ` C ) |
2 |
1
|
mrcfval |
|- ( C e. ( Moore ` X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) ) |
3 |
2
|
adantr |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> F = ( x e. ~P X |-> |^| { s e. C | x C_ s } ) ) |
4 |
|
sseq1 |
|- ( x = U -> ( x C_ s <-> U C_ s ) ) |
5 |
4
|
rabbidv |
|- ( x = U -> { s e. C | x C_ s } = { s e. C | U C_ s } ) |
6 |
5
|
inteqd |
|- ( x = U -> |^| { s e. C | x C_ s } = |^| { s e. C | U C_ s } ) |
7 |
6
|
adantl |
|- ( ( ( C e. ( Moore ` X ) /\ U C_ X ) /\ x = U ) -> |^| { s e. C | x C_ s } = |^| { s e. C | U C_ s } ) |
8 |
|
mre1cl |
|- ( C e. ( Moore ` X ) -> X e. C ) |
9 |
|
elpw2g |
|- ( X e. C -> ( U e. ~P X <-> U C_ X ) ) |
10 |
8 9
|
syl |
|- ( C e. ( Moore ` X ) -> ( U e. ~P X <-> U C_ X ) ) |
11 |
10
|
biimpar |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> U e. ~P X ) |
12 |
|
sseq2 |
|- ( s = X -> ( U C_ s <-> U C_ X ) ) |
13 |
8
|
adantr |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> X e. C ) |
14 |
|
simpr |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> U C_ X ) |
15 |
12 13 14
|
elrabd |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> X e. { s e. C | U C_ s } ) |
16 |
15
|
ne0d |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> { s e. C | U C_ s } =/= (/) ) |
17 |
|
intex |
|- ( { s e. C | U C_ s } =/= (/) <-> |^| { s e. C | U C_ s } e. _V ) |
18 |
16 17
|
sylib |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> |^| { s e. C | U C_ s } e. _V ) |
19 |
3 7 11 18
|
fvmptd |
|- ( ( C e. ( Moore ` X ) /\ U C_ X ) -> ( F ` U ) = |^| { s e. C | U C_ s } ) |