Step |
Hyp |
Ref |
Expression |
1 |
|
elfvex |
⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → 𝑋 ∈ V ) |
2 |
|
elfvex |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → 𝑋 ∈ V ) |
3 |
2
|
adantr |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) → 𝑋 ∈ V ) |
4 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( Moore ‘ 𝑥 ) = ( Moore ‘ 𝑋 ) ) |
5 |
|
pweq |
⊢ ( 𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋 ) |
6 |
5 5
|
feq23d |
⊢ ( 𝑥 = 𝑋 → ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ↔ 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ) ) |
7 |
5
|
raleqdv |
⊢ ( 𝑥 = 𝑋 → ( ∀ 𝑠 ∈ 𝒫 𝑥 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ↔ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) |
8 |
6 7
|
anbi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑠 ∈ 𝒫 𝑥 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ↔ ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |
9 |
8
|
exbidv |
⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑓 ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑠 ∈ 𝒫 𝑥 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ↔ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |
10 |
4 9
|
rabeqbidv |
⊢ ( 𝑥 = 𝑋 → { 𝑐 ∈ ( Moore ‘ 𝑥 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑠 ∈ 𝒫 𝑥 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } = { 𝑐 ∈ ( Moore ‘ 𝑋 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ) |
11 |
|
df-acs |
⊢ ACS = ( 𝑥 ∈ V ↦ { 𝑐 ∈ ( Moore ‘ 𝑥 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑠 ∈ 𝒫 𝑥 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ) |
12 |
|
fvex |
⊢ ( Moore ‘ 𝑋 ) ∈ V |
13 |
12
|
rabex |
⊢ { 𝑐 ∈ ( Moore ‘ 𝑋 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ∈ V |
14 |
10 11 13
|
fvmpt |
⊢ ( 𝑋 ∈ V → ( ACS ‘ 𝑋 ) = { 𝑐 ∈ ( Moore ‘ 𝑋 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ) |
15 |
14
|
eleq2d |
⊢ ( 𝑋 ∈ V → ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ↔ 𝐶 ∈ { 𝑐 ∈ ( Moore ‘ 𝑋 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ) ) |
16 |
|
eleq2 |
⊢ ( 𝑐 = 𝐶 → ( 𝑠 ∈ 𝑐 ↔ 𝑠 ∈ 𝐶 ) ) |
17 |
16
|
bibi1d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ↔ ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) |
18 |
17
|
ralbidv |
⊢ ( 𝑐 = 𝐶 → ( ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ↔ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) |
19 |
18
|
anbi2d |
⊢ ( 𝑐 = 𝐶 → ( ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ↔ ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |
20 |
19
|
exbidv |
⊢ ( 𝑐 = 𝐶 → ( ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ↔ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |
21 |
20
|
elrab |
⊢ ( 𝐶 ∈ { 𝑐 ∈ ( Moore ‘ 𝑋 ) ∣ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝑐 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) } ↔ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |
22 |
15 21
|
bitrdi |
⊢ ( 𝑋 ∈ V → ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) ) |
23 |
1 3 22
|
pm5.21nii |
⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑋 ⟶ 𝒫 𝑋 ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝑠 ∈ 𝐶 ↔ ∪ ( 𝑓 “ ( 𝒫 𝑠 ∩ Fin ) ) ⊆ 𝑠 ) ) ) ) |