Step |
Hyp |
Ref |
Expression |
1 |
|
simplr |
|- ( ( ( A e. On /\ L < L < |
2 |
|
ssltex1 |
|- ( L < L e. _V ) |
3 |
1 2
|
syl |
|- ( ( ( A e. On /\ L < L e. _V ) |
4 |
|
simprl |
|- ( ( ( A e. On /\ L < L C_ ( _Old ` A ) ) |
5 |
3 4
|
elpwd |
|- ( ( ( A e. On /\ L < L e. ~P ( _Old ` A ) ) |
6 |
|
ssltex2 |
|- ( L < R e. _V ) |
7 |
1 6
|
syl |
|- ( ( ( A e. On /\ L < R e. _V ) |
8 |
|
simprr |
|- ( ( ( A e. On /\ L < R C_ ( _Old ` A ) ) |
9 |
7 8
|
elpwd |
|- ( ( ( A e. On /\ L < R e. ~P ( _Old ` A ) ) |
10 |
|
eqidd |
|- ( ( ( A e. On /\ L < ( L |s R ) = ( L |s R ) ) |
11 |
|
breq1 |
|- ( l = L -> ( l < L < |
12 |
|
oveq1 |
|- ( l = L -> ( l |s r ) = ( L |s r ) ) |
13 |
12
|
eqeq1d |
|- ( l = L -> ( ( l |s r ) = ( L |s R ) <-> ( L |s r ) = ( L |s R ) ) ) |
14 |
11 13
|
anbi12d |
|- ( l = L -> ( ( l < ( L < |
15 |
|
breq2 |
|- ( r = R -> ( L < L < |
16 |
|
oveq2 |
|- ( r = R -> ( L |s r ) = ( L |s R ) ) |
17 |
16
|
eqeq1d |
|- ( r = R -> ( ( L |s r ) = ( L |s R ) <-> ( L |s R ) = ( L |s R ) ) ) |
18 |
15 17
|
anbi12d |
|- ( r = R -> ( ( L < ( L < |
19 |
14 18
|
rspc2ev |
|- ( ( L e. ~P ( _Old ` A ) /\ R e. ~P ( _Old ` A ) /\ ( L < E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l < |
20 |
5 9 1 10 19
|
syl112anc |
|- ( ( ( A e. On /\ L < E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l < |
21 |
|
elmade2 |
|- ( A e. On -> ( ( L |s R ) e. ( _M ` A ) <-> E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l < |
22 |
21
|
ad2antrr |
|- ( ( ( A e. On /\ L < ( ( L |s R ) e. ( _M ` A ) <-> E. l e. ~P ( _Old ` A ) E. r e. ~P ( _Old ` A ) ( l < |
23 |
20 22
|
mpbird |
|- ( ( ( A e. On /\ L < ( L |s R ) e. ( _M ` A ) ) |