Step |
Hyp |
Ref |
Expression |
1 |
|
inss1 |
|- ( S i^i T ) C_ S |
2 |
|
simpl |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> x e. ( S i^i T ) ) |
3 |
2
|
elin2d |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> x e. T ) |
4 |
|
simpr |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> y e. ( S i^i T ) ) |
5 |
4
|
elin2d |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> y e. T ) |
6 |
3 5
|
ovresd |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> ( x ( H |` ( T X. T ) ) y ) = ( x H y ) ) |
7 |
|
eqimss |
|- ( ( x ( H |` ( T X. T ) ) y ) = ( x H y ) -> ( x ( H |` ( T X. T ) ) y ) C_ ( x H y ) ) |
8 |
6 7
|
syl |
|- ( ( x e. ( S i^i T ) /\ y e. ( S i^i T ) ) -> ( x ( H |` ( T X. T ) ) y ) C_ ( x H y ) ) |
9 |
8
|
rgen2 |
|- A. x e. ( S i^i T ) A. y e. ( S i^i T ) ( x ( H |` ( T X. T ) ) y ) C_ ( x H y ) |
10 |
1 9
|
pm3.2i |
|- ( ( S i^i T ) C_ S /\ A. x e. ( S i^i T ) A. y e. ( S i^i T ) ( x ( H |` ( T X. T ) ) y ) C_ ( x H y ) ) |
11 |
|
simpl |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> H Fn ( S X. S ) ) |
12 |
|
inss1 |
|- ( ( S X. S ) i^i ( T X. T ) ) C_ ( S X. S ) |
13 |
|
fnssres |
|- ( ( H Fn ( S X. S ) /\ ( ( S X. S ) i^i ( T X. T ) ) C_ ( S X. S ) ) -> ( H |` ( ( S X. S ) i^i ( T X. T ) ) ) Fn ( ( S X. S ) i^i ( T X. T ) ) ) |
14 |
11 12 13
|
sylancl |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( H |` ( ( S X. S ) i^i ( T X. T ) ) ) Fn ( ( S X. S ) i^i ( T X. T ) ) ) |
15 |
|
resres |
|- ( ( H |` ( S X. S ) ) |` ( T X. T ) ) = ( H |` ( ( S X. S ) i^i ( T X. T ) ) ) |
16 |
|
fnresdm |
|- ( H Fn ( S X. S ) -> ( H |` ( S X. S ) ) = H ) |
17 |
16
|
adantr |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( H |` ( S X. S ) ) = H ) |
18 |
17
|
reseq1d |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( ( H |` ( S X. S ) ) |` ( T X. T ) ) = ( H |` ( T X. T ) ) ) |
19 |
15 18
|
eqtr3id |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( H |` ( ( S X. S ) i^i ( T X. T ) ) ) = ( H |` ( T X. T ) ) ) |
20 |
|
inxp |
|- ( ( S X. S ) i^i ( T X. T ) ) = ( ( S i^i T ) X. ( S i^i T ) ) |
21 |
20
|
a1i |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( ( S X. S ) i^i ( T X. T ) ) = ( ( S i^i T ) X. ( S i^i T ) ) ) |
22 |
19 21
|
fneq12d |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( ( H |` ( ( S X. S ) i^i ( T X. T ) ) ) Fn ( ( S X. S ) i^i ( T X. T ) ) <-> ( H |` ( T X. T ) ) Fn ( ( S i^i T ) X. ( S i^i T ) ) ) ) |
23 |
14 22
|
mpbid |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( H |` ( T X. T ) ) Fn ( ( S i^i T ) X. ( S i^i T ) ) ) |
24 |
|
simpr |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> S e. V ) |
25 |
23 11 24
|
isssc |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( ( H |` ( T X. T ) ) C_cat H <-> ( ( S i^i T ) C_ S /\ A. x e. ( S i^i T ) A. y e. ( S i^i T ) ( x ( H |` ( T X. T ) ) y ) C_ ( x H y ) ) ) ) |
26 |
10 25
|
mpbiri |
|- ( ( H Fn ( S X. S ) /\ S e. V ) -> ( H |` ( T X. T ) ) C_cat H ) |