Step |
Hyp |
Ref |
Expression |
1 |
|
sseq1 |
|- ( a = c -> ( a C_ b <-> c C_ b ) ) |
2 |
|
fveq2 |
|- ( a = c -> ( F ` a ) = ( F ` c ) ) |
3 |
2
|
sseq1d |
|- ( a = c -> ( ( F ` a ) C_ ( F ` b ) <-> ( F ` c ) C_ ( F ` b ) ) ) |
4 |
1 3
|
imbi12d |
|- ( a = c -> ( ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) <-> ( c C_ b -> ( F ` c ) C_ ( F ` b ) ) ) ) |
5 |
|
sseq2 |
|- ( b = d -> ( c C_ b <-> c C_ d ) ) |
6 |
|
fveq2 |
|- ( b = d -> ( F ` b ) = ( F ` d ) ) |
7 |
6
|
sseq2d |
|- ( b = d -> ( ( F ` c ) C_ ( F ` b ) <-> ( F ` c ) C_ ( F ` d ) ) ) |
8 |
5 7
|
imbi12d |
|- ( b = d -> ( ( c C_ b -> ( F ` c ) C_ ( F ` b ) ) <-> ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) ) |
9 |
4 8
|
cbvral2vw |
|- ( A. a e. ~P A A. b e. ~P A ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) <-> A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) |
10 |
|
inss1 |
|- ( a i^i b ) C_ a |
11 |
|
inss2 |
|- ( a i^i b ) C_ b |
12 |
|
elpwi |
|- ( b e. ~P A -> b C_ A ) |
13 |
11 12
|
sstrid |
|- ( b e. ~P A -> ( a i^i b ) C_ A ) |
14 |
|
vex |
|- b e. _V |
15 |
14
|
inex2 |
|- ( a i^i b ) e. _V |
16 |
15
|
elpw |
|- ( ( a i^i b ) e. ~P A <-> ( a i^i b ) C_ A ) |
17 |
13 16
|
sylibr |
|- ( b e. ~P A -> ( a i^i b ) e. ~P A ) |
18 |
17
|
ad2antll |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( a i^i b ) e. ~P A ) |
19 |
|
simprl |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> a e. ~P A ) |
20 |
|
simpl |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) |
21 |
|
sseq1 |
|- ( c = ( a i^i b ) -> ( c C_ d <-> ( a i^i b ) C_ d ) ) |
22 |
|
fveq2 |
|- ( c = ( a i^i b ) -> ( F ` c ) = ( F ` ( a i^i b ) ) ) |
23 |
22
|
sseq1d |
|- ( c = ( a i^i b ) -> ( ( F ` c ) C_ ( F ` d ) <-> ( F ` ( a i^i b ) ) C_ ( F ` d ) ) ) |
24 |
21 23
|
imbi12d |
|- ( c = ( a i^i b ) -> ( ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) <-> ( ( a i^i b ) C_ d -> ( F ` ( a i^i b ) ) C_ ( F ` d ) ) ) ) |
25 |
|
sseq2 |
|- ( d = a -> ( ( a i^i b ) C_ d <-> ( a i^i b ) C_ a ) ) |
26 |
|
fveq2 |
|- ( d = a -> ( F ` d ) = ( F ` a ) ) |
27 |
26
|
sseq2d |
|- ( d = a -> ( ( F ` ( a i^i b ) ) C_ ( F ` d ) <-> ( F ` ( a i^i b ) ) C_ ( F ` a ) ) ) |
28 |
25 27
|
imbi12d |
|- ( d = a -> ( ( ( a i^i b ) C_ d -> ( F ` ( a i^i b ) ) C_ ( F ` d ) ) <-> ( ( a i^i b ) C_ a -> ( F ` ( a i^i b ) ) C_ ( F ` a ) ) ) ) |
29 |
24 28
|
rspc2va |
|- ( ( ( ( a i^i b ) e. ~P A /\ a e. ~P A ) /\ A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) -> ( ( a i^i b ) C_ a -> ( F ` ( a i^i b ) ) C_ ( F ` a ) ) ) |
30 |
18 19 20 29
|
syl21anc |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( a i^i b ) C_ a -> ( F ` ( a i^i b ) ) C_ ( F ` a ) ) ) |
31 |
10 30
|
mpi |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` ( a i^i b ) ) C_ ( F ` a ) ) |
32 |
|
simprr |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> b e. ~P A ) |
33 |
|
sseq2 |
|- ( d = b -> ( ( a i^i b ) C_ d <-> ( a i^i b ) C_ b ) ) |
34 |
|
fveq2 |
|- ( d = b -> ( F ` d ) = ( F ` b ) ) |
35 |
34
|
sseq2d |
|- ( d = b -> ( ( F ` ( a i^i b ) ) C_ ( F ` d ) <-> ( F ` ( a i^i b ) ) C_ ( F ` b ) ) ) |
36 |
33 35
|
imbi12d |
|- ( d = b -> ( ( ( a i^i b ) C_ d -> ( F ` ( a i^i b ) ) C_ ( F ` d ) ) <-> ( ( a i^i b ) C_ b -> ( F ` ( a i^i b ) ) C_ ( F ` b ) ) ) ) |
37 |
24 36
|
rspc2va |
|- ( ( ( ( a i^i b ) e. ~P A /\ b e. ~P A ) /\ A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) -> ( ( a i^i b ) C_ b -> ( F ` ( a i^i b ) ) C_ ( F ` b ) ) ) |
38 |
18 32 20 37
|
syl21anc |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( ( a i^i b ) C_ b -> ( F ` ( a i^i b ) ) C_ ( F ` b ) ) ) |
39 |
11 38
|
mpi |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` ( a i^i b ) ) C_ ( F ` b ) ) |
40 |
31 39
|
ssind |
|- ( ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) /\ ( a e. ~P A /\ b e. ~P A ) ) -> ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) ) |
41 |
40
|
ralrimivva |
|- ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) -> A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) ) |
42 |
|
dfss |
|- ( c C_ d <-> c = ( c i^i d ) ) |
43 |
|
fveq2 |
|- ( c = ( c i^i d ) -> ( F ` c ) = ( F ` ( c i^i d ) ) ) |
44 |
43
|
adantl |
|- ( ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) /\ c = ( c i^i d ) ) -> ( F ` c ) = ( F ` ( c i^i d ) ) ) |
45 |
|
ineq1 |
|- ( a = c -> ( a i^i b ) = ( c i^i b ) ) |
46 |
45
|
fveq2d |
|- ( a = c -> ( F ` ( a i^i b ) ) = ( F ` ( c i^i b ) ) ) |
47 |
2
|
ineq1d |
|- ( a = c -> ( ( F ` a ) i^i ( F ` b ) ) = ( ( F ` c ) i^i ( F ` b ) ) ) |
48 |
46 47
|
sseq12d |
|- ( a = c -> ( ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) <-> ( F ` ( c i^i b ) ) C_ ( ( F ` c ) i^i ( F ` b ) ) ) ) |
49 |
|
ineq2 |
|- ( b = d -> ( c i^i b ) = ( c i^i d ) ) |
50 |
49
|
fveq2d |
|- ( b = d -> ( F ` ( c i^i b ) ) = ( F ` ( c i^i d ) ) ) |
51 |
6
|
ineq2d |
|- ( b = d -> ( ( F ` c ) i^i ( F ` b ) ) = ( ( F ` c ) i^i ( F ` d ) ) ) |
52 |
50 51
|
sseq12d |
|- ( b = d -> ( ( F ` ( c i^i b ) ) C_ ( ( F ` c ) i^i ( F ` b ) ) <-> ( F ` ( c i^i d ) ) C_ ( ( F ` c ) i^i ( F ` d ) ) ) ) |
53 |
48 52
|
rspc2va |
|- ( ( ( c e. ~P A /\ d e. ~P A ) /\ A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) ) -> ( F ` ( c i^i d ) ) C_ ( ( F ` c ) i^i ( F ` d ) ) ) |
54 |
53
|
ancoms |
|- ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) -> ( F ` ( c i^i d ) ) C_ ( ( F ` c ) i^i ( F ` d ) ) ) |
55 |
|
inss2 |
|- ( ( F ` c ) i^i ( F ` d ) ) C_ ( F ` d ) |
56 |
54 55
|
sstrdi |
|- ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) -> ( F ` ( c i^i d ) ) C_ ( F ` d ) ) |
57 |
56
|
adantr |
|- ( ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) /\ c = ( c i^i d ) ) -> ( F ` ( c i^i d ) ) C_ ( F ` d ) ) |
58 |
44 57
|
eqsstrd |
|- ( ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) /\ c = ( c i^i d ) ) -> ( F ` c ) C_ ( F ` d ) ) |
59 |
58
|
ex |
|- ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) -> ( c = ( c i^i d ) -> ( F ` c ) C_ ( F ` d ) ) ) |
60 |
42 59
|
syl5bi |
|- ( ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) /\ ( c e. ~P A /\ d e. ~P A ) ) -> ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) |
61 |
60
|
ralrimivva |
|- ( A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) -> A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) ) |
62 |
41 61
|
impbii |
|- ( A. c e. ~P A A. d e. ~P A ( c C_ d -> ( F ` c ) C_ ( F ` d ) ) <-> A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) ) |
63 |
9 62
|
bitri |
|- ( A. a e. ~P A A. b e. ~P A ( a C_ b -> ( F ` a ) C_ ( F ` b ) ) <-> A. a e. ~P A A. b e. ~P A ( F ` ( a i^i b ) ) C_ ( ( F ` a ) i^i ( F ` b ) ) ) |