Proof of Theorem acsfn2
Step | Hyp | Ref
| Expression |
1 | | elpwi 4548 |
. . . . 5
⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) |
2 | | ralss 3996 |
. . . . . 6
⊢ (𝑎 ⊆ 𝑋 → (∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ 𝑋 (𝑏 ∈ 𝑎 → ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎))) |
3 | | ralss 3996 |
. . . . . . . 8
⊢ (𝑎 ⊆ 𝑋 → (∀𝑐 ∈ 𝑎 (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝑋 (𝑐 ∈ 𝑎 → (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎)))) |
4 | | r19.21v 3103 |
. . . . . . . 8
⊢
(∀𝑐 ∈
𝑎 (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎) ↔ (𝑏 ∈ 𝑎 → ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎)) |
5 | | impexp 451 |
. . . . . . . . . 10
⊢ (((𝑐 ∈ 𝑎 ∧ 𝑏 ∈ 𝑎) → 𝐸 ∈ 𝑎) ↔ (𝑐 ∈ 𝑎 → (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎))) |
6 | | vex 3435 |
. . . . . . . . . . . 12
⊢ 𝑐 ∈ V |
7 | | vex 3435 |
. . . . . . . . . . . 12
⊢ 𝑏 ∈ V |
8 | 6, 7 | prss 4759 |
. . . . . . . . . . 11
⊢ ((𝑐 ∈ 𝑎 ∧ 𝑏 ∈ 𝑎) ↔ {𝑐, 𝑏} ⊆ 𝑎) |
9 | 8 | imbi1i 350 |
. . . . . . . . . 10
⊢ (((𝑐 ∈ 𝑎 ∧ 𝑏 ∈ 𝑎) → 𝐸 ∈ 𝑎) ↔ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)) |
10 | 5, 9 | bitr3i 276 |
. . . . . . . . 9
⊢ ((𝑐 ∈ 𝑎 → (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎)) ↔ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)) |
11 | 10 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑐 ∈
𝑋 (𝑐 ∈ 𝑎 → (𝑏 ∈ 𝑎 → 𝐸 ∈ 𝑎)) ↔ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)) |
12 | 3, 4, 11 | 3bitr3g 313 |
. . . . . . 7
⊢ (𝑎 ⊆ 𝑋 → ((𝑏 ∈ 𝑎 → ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎))) |
13 | 12 | ralbidv 3123 |
. . . . . 6
⊢ (𝑎 ⊆ 𝑋 → (∀𝑏 ∈ 𝑋 (𝑏 ∈ 𝑎 → ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎))) |
14 | 2, 13 | bitrd 278 |
. . . . 5
⊢ (𝑎 ⊆ 𝑋 → (∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎))) |
15 | 1, 14 | syl 17 |
. . . 4
⊢ (𝑎 ∈ 𝒫 𝑋 → (∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎))) |
16 | 15 | rabbiia 3405 |
. . 3
⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} |
17 | | riinrab 5018 |
. . 3
⊢
(𝒫 𝑋 ∩
∩ 𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} |
18 | 16, 17 | eqtr4i 2771 |
. 2
⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} = (𝒫 𝑋 ∩ ∩
𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) |
19 | | mreacs 17365 |
. . 3
⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)) |
20 | | riinrab 5018 |
. . . . . . 7
⊢
(𝒫 𝑋 ∩
∩ 𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} |
21 | 19 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋)) |
22 | | simpll 764 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → 𝑋 ∈ 𝑉) |
23 | | simprr 770 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → 𝐸 ∈ 𝑋) |
24 | | prssi 4760 |
. . . . . . . . . . . . . 14
⊢ ((𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → {𝑐, 𝑏} ⊆ 𝑋) |
25 | 24 | ancoms 459 |
. . . . . . . . . . . . 13
⊢ ((𝑏 ∈ 𝑋 ∧ 𝑐 ∈ 𝑋) → {𝑐, 𝑏} ⊆ 𝑋) |
26 | 25 | ad2ant2lr 745 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → {𝑐, 𝑏} ⊆ 𝑋) |
27 | | prfi 9067 |
. . . . . . . . . . . . 13
⊢ {𝑐, 𝑏} ∈ Fin |
28 | 27 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → {𝑐, 𝑏} ∈ Fin) |
29 | | acsfn 17366 |
. . . . . . . . . . . 12
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐸 ∈ 𝑋) ∧ ({𝑐, 𝑏} ⊆ 𝑋 ∧ {𝑐, 𝑏} ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) |
30 | 22, 23, 26, 28, 29 | syl22anc 836 |
. . . . . . . . . . 11
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝐸 ∈ 𝑋)) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) |
31 | 30 | expr 457 |
. . . . . . . . . 10
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ 𝑐 ∈ 𝑋) → (𝐸 ∈ 𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))) |
32 | 31 | ralimdva 3105 |
. . . . . . . . 9
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) → (∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋 → ∀𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))) |
33 | 32 | imp 407 |
. . . . . . . 8
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → ∀𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) |
34 | | mreriincl 17305 |
. . . . . . . 8
⊢
(((ACS‘𝑋)
∈ (Moore‘𝒫 𝑋) ∧ ∀𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) → (𝒫 𝑋 ∩ ∩
𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋)) |
35 | 21, 33, 34 | syl2anc 584 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → (𝒫 𝑋 ∩ ∩
𝑐 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋)) |
36 | 20, 35 | eqeltrrid 2846 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) |
37 | 36 | ex 413 |
. . . . 5
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ 𝑋) → (∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))) |
38 | 37 | ralimdva 3105 |
. . . 4
⊢ (𝑋 ∈ 𝑉 → (∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋 → ∀𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))) |
39 | 38 | imp 407 |
. . 3
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → ∀𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) |
40 | | mreriincl 17305 |
. . 3
⊢
(((ACS‘𝑋)
∈ (Moore‘𝒫 𝑋) ∧ ∀𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) → (𝒫 𝑋 ∩ ∩
𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋)) |
41 | 19, 39, 40 | syl2an2r 682 |
. 2
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → (𝒫 𝑋 ∩ ∩
𝑏 ∈ 𝑋 {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑐 ∈ 𝑋 ({𝑐, 𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋)) |
42 | 18, 41 | eqeltrid 2845 |
1
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ 𝑎 ∀𝑐 ∈ 𝑎 𝐸 ∈ 𝑎} ∈ (ACS‘𝑋)) |