Step |
Hyp |
Ref |
Expression |
1 |
|
ssrab2 |
⊢ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ⊆ 𝒫 𝑋 |
2 |
1
|
a1i |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ⊆ 𝒫 𝑋 ) |
3 |
|
pweq |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → 𝒫 𝑠 = 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ) |
4 |
3
|
ineq1d |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → ( 𝒫 𝑠 ∩ Fin ) = ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) |
5 |
4
|
imaeq2d |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ) |
6 |
5
|
unieqd |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ) |
7 |
|
id |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) ) |
8 |
6 7
|
sseq12d |
⊢ ( 𝑠 = ( 𝑋 ∩ ∩ 𝑡 ) → ( ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ↔ ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ( 𝑋 ∩ ∩ 𝑡 ) ) ) |
9 |
|
inss1 |
⊢ ( 𝑋 ∩ ∩ 𝑡 ) ⊆ 𝑋 |
10 |
|
elpw2g |
⊢ ( 𝑋 ∈ 𝑉 → ( ( 𝑋 ∩ ∩ 𝑡 ) ∈ 𝒫 𝑋 ↔ ( 𝑋 ∩ ∩ 𝑡 ) ⊆ 𝑋 ) ) |
11 |
9 10
|
mpbiri |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝑋 ∩ ∩ 𝑡 ) ∈ 𝒫 𝑋 ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ( 𝑋 ∩ ∩ 𝑡 ) ∈ 𝒫 𝑋 ) |
13 |
|
imassrn |
⊢ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ran 𝐹 |
14 |
|
frn |
⊢ ( 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋 ) |
15 |
14
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ran 𝐹 ⊆ 𝒫 𝑋 ) |
16 |
13 15
|
sstrid |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝒫 𝑋 ) |
17 |
16
|
unissd |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ∪ 𝒫 𝑋 ) |
18 |
|
unipw |
⊢ ∪ 𝒫 𝑋 = 𝑋 |
19 |
17 18
|
sseqtrdi |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝑋 ) |
20 |
19
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝑋 ) |
21 |
|
inss2 |
⊢ ( 𝑋 ∩ ∩ 𝑡 ) ⊆ ∩ 𝑡 |
22 |
|
intss1 |
⊢ ( 𝑎 ∈ 𝑡 → ∩ 𝑡 ⊆ 𝑎 ) |
23 |
21 22
|
sstrid |
⊢ ( 𝑎 ∈ 𝑡 → ( 𝑋 ∩ ∩ 𝑡 ) ⊆ 𝑎 ) |
24 |
23
|
adantl |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ( 𝑋 ∩ ∩ 𝑡 ) ⊆ 𝑎 ) |
25 |
24
|
sspwd |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ⊆ 𝒫 𝑎 ) |
26 |
25
|
ssrind |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ⊆ ( 𝒫 𝑎 ∩ Fin ) ) |
27 |
|
imass2 |
⊢ ( ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ⊆ ( 𝒫 𝑎 ∩ Fin ) → ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ) |
28 |
26 27
|
syl |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ) |
29 |
28
|
unissd |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ) |
30 |
|
ssel2 |
⊢ ( ( 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∧ 𝑎 ∈ 𝑡 ) → 𝑎 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) |
31 |
|
pweq |
⊢ ( 𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎 ) |
32 |
31
|
ineq1d |
⊢ ( 𝑠 = 𝑎 → ( 𝒫 𝑠 ∩ Fin ) = ( 𝒫 𝑎 ∩ Fin ) ) |
33 |
32
|
imaeq2d |
⊢ ( 𝑠 = 𝑎 → ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ) |
34 |
33
|
unieqd |
⊢ ( 𝑠 = 𝑎 → ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ) |
35 |
|
id |
⊢ ( 𝑠 = 𝑎 → 𝑠 = 𝑎 ) |
36 |
34 35
|
sseq12d |
⊢ ( 𝑠 = 𝑎 → ( ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ↔ ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ⊆ 𝑎 ) ) |
37 |
36
|
elrab |
⊢ ( 𝑎 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ( 𝑎 ∈ 𝒫 𝑋 ∧ ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ⊆ 𝑎 ) ) |
38 |
37
|
simprbi |
⊢ ( 𝑎 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } → ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ⊆ 𝑎 ) |
39 |
30 38
|
syl |
⊢ ( ( 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∧ 𝑎 ∈ 𝑡 ) → ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ⊆ 𝑎 ) |
40 |
39
|
adantll |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ∪ ( 𝐹 “ ( 𝒫 𝑎 ∩ Fin ) ) ⊆ 𝑎 ) |
41 |
29 40
|
sstrd |
⊢ ( ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) ∧ 𝑎 ∈ 𝑡 ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝑎 ) |
42 |
41
|
ralrimiva |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ∀ 𝑎 ∈ 𝑡 ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝑎 ) |
43 |
|
ssint |
⊢ ( ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ∩ 𝑡 ↔ ∀ 𝑎 ∈ 𝑡 ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ 𝑎 ) |
44 |
42 43
|
sylibr |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ∩ 𝑡 ) |
45 |
20 44
|
ssind |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ∪ ( 𝐹 “ ( 𝒫 ( 𝑋 ∩ ∩ 𝑡 ) ∩ Fin ) ) ⊆ ( 𝑋 ∩ ∩ 𝑡 ) ) |
46 |
8 12 45
|
elrabd |
⊢ ( ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ∧ 𝑡 ⊆ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) → ( 𝑋 ∩ ∩ 𝑡 ) ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ) |
47 |
2 46
|
ismred2 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∈ ( Moore ‘ 𝑋 ) ) |
48 |
|
fssxp |
⊢ ( 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 → 𝐹 ⊆ ( 𝒫 𝑋 × 𝒫 𝑋 ) ) |
49 |
|
pwexg |
⊢ ( 𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V ) |
50 |
49 49
|
xpexd |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝒫 𝑋 × 𝒫 𝑋 ) ∈ V ) |
51 |
|
ssexg |
⊢ ( ( 𝐹 ⊆ ( 𝒫 𝑋 × 𝒫 𝑋 ) ∧ ( 𝒫 𝑋 × 𝒫 𝑋 ) ∈ V ) → 𝐹 ∈ V ) |
52 |
48 50 51
|
syl2anr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → 𝐹 ∈ V ) |
53 |
|
simpr |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) |
54 |
|
pweq |
⊢ ( 𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡 ) |
55 |
54
|
ineq1d |
⊢ ( 𝑠 = 𝑡 → ( 𝒫 𝑠 ∩ Fin ) = ( 𝒫 𝑡 ∩ Fin ) ) |
56 |
55
|
imaeq2d |
⊢ ( 𝑠 = 𝑡 → ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
57 |
56
|
unieqd |
⊢ ( 𝑠 = 𝑡 → ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
58 |
|
id |
⊢ ( 𝑠 = 𝑡 → 𝑠 = 𝑡 ) |
59 |
57 58
|
sseq12d |
⊢ ( 𝑠 = 𝑡 → ( ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) |
60 |
59
|
elrab3 |
⊢ ( 𝑡 ∈ 𝒫 𝑋 → ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) |
61 |
60
|
rgen |
⊢ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) |
62 |
53 61
|
jctir |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ( 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) |
63 |
|
feq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ↔ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ) |
64 |
|
imaeq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) = ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
65 |
64
|
unieqd |
⊢ ( 𝑓 = 𝐹 → ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) = ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
66 |
65
|
sseq1d |
⊢ ( 𝑓 = 𝐹 → ( ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) |
67 |
66
|
bibi2d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ↔ ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) |
68 |
67
|
ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ↔ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) |
69 |
63 68
|
anbi12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ↔ ( 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝐹 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) ) |
70 |
52 62 69
|
spcedv |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) |
71 |
|
isacs |
⊢ ( { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∈ ( ACS ‘ 𝑋 ) ↔ ( { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∈ ( Moore ‘ 𝑋 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑡 ∈ 𝒫 𝑋 ( 𝑡 ∈ { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ↔ ∪ ( 𝑓 “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝑡 ) ) ) ) |
72 |
47 70 71
|
sylanbrc |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐹 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) → { 𝑠 ∈ 𝒫 𝑋 ∣ ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 } ∈ ( ACS ‘ 𝑋 ) ) |