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