Step | Hyp | Ref
| Expression |
1 | | funmpt 5840 |
. . . . . . 7
⊢ Fun
(𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) |
2 | | funiunfv 6410 |
. . . . . . 7
⊢ (Fun
(𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) → ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin))) |
3 | 1, 2 | mp1i 13 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪
𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin))) |
4 | | inss1 3795 |
. . . . . . . . . . . . 13
⊢
(𝒫 𝑎 ∩
Fin) ⊆ 𝒫 𝑎 |
5 | 4 | sseli 3564 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎) |
6 | 5 | elpwid 4118 |
. . . . . . . . . . 11
⊢ (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ⊆ 𝑎) |
7 | | elpwi 4117 |
. . . . . . . . . . 11
⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋) |
8 | 6, 7 | sylan9ssr 3582 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ⊆ 𝑋) |
9 | | selpw 4115 |
. . . . . . . . . 10
⊢ (𝑐 ∈ 𝒫 𝑋 ↔ 𝑐 ⊆ 𝑋) |
10 | 8, 9 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝑎 ∈ 𝒫 𝑋 ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋) |
11 | 10 | adantll 746 |
. . . . . . . 8
⊢
(((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋) |
12 | | eqeq1 2614 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑐 → (𝑏 = 𝑇 ↔ 𝑐 = 𝑇)) |
13 | 12 | ifbid 4058 |
. . . . . . . . 9
⊢ (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅)) |
14 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) |
15 | | snex 4835 |
. . . . . . . . . 10
⊢ {𝐾} ∈ V |
16 | | 0ex 4718 |
. . . . . . . . . 10
⊢ ∅
∈ V |
17 | 15, 16 | ifex 4106 |
. . . . . . . . 9
⊢ if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V |
18 | 13, 14, 17 | fvmpt 6191 |
. . . . . . . 8
⊢ (𝑐 ∈ 𝒫 𝑋 → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅)) |
19 | 11, 18 | syl 17 |
. . . . . . 7
⊢
(((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅)) |
20 | 19 | iuneq2dv 4478 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪
𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅)) |
21 | 3, 20 | eqtr3d 2646 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ∪
((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) = ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅)) |
22 | 21 | sseq1d 3595 |
. . . 4
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∪
((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎 ↔ ∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)) |
23 | | iunss 4497 |
. . . . 5
⊢ (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎) |
24 | | sseq1 3589 |
. . . . . . . . 9
⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)) |
25 | 24 | bibi1d 332 |
. . . . . . . 8
⊢ ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))) |
26 | | sseq1 3589 |
. . . . . . . . 9
⊢ (∅
= if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)) |
27 | 26 | bibi1d 332 |
. . . . . . . 8
⊢ (∅
= if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))) |
28 | | snssg 4268 |
. . . . . . . . . 10
⊢ (𝐾 ∈ 𝑋 → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎)) |
29 | 28 | adantr 480 |
. . . . . . . . 9
⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ {𝐾} ⊆ 𝑎)) |
30 | | biimt 349 |
. . . . . . . . . 10
⊢ (𝑐 = 𝑇 → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
31 | 30 | adantl 481 |
. . . . . . . . 9
⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → (𝐾 ∈ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
32 | 29, 31 | bitr3d 269 |
. . . . . . . 8
⊢ ((𝐾 ∈ 𝑋 ∧ 𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
33 | | 0ss 3924 |
. . . . . . . . . . 11
⊢ ∅
⊆ 𝑎 |
34 | 33 | a1i 11 |
. . . . . . . . . 10
⊢ (¬
𝑐 = 𝑇 → ∅ ⊆ 𝑎) |
35 | | pm2.21 119 |
. . . . . . . . . 10
⊢ (¬
𝑐 = 𝑇 → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) |
36 | 34, 35 | 2thd 254 |
. . . . . . . . 9
⊢ (¬
𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
37 | 36 | adantl 481 |
. . . . . . . 8
⊢ ((𝐾 ∈ 𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
38 | 25, 27, 32, 37 | ifbothda 4073 |
. . . . . . 7
⊢ (𝐾 ∈ 𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
39 | 38 | ralbidv 2969 |
. . . . . 6
⊢ (𝐾 ∈ 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
40 | 39 | ad3antlr 763 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
41 | 23, 40 | syl5bb 271 |
. . . 4
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∪ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎))) |
42 | | sspwb 4844 |
. . . . . . . . 9
⊢ (𝑎 ⊆ 𝑋 ↔ 𝒫 𝑎 ⊆ 𝒫 𝑋) |
43 | 7, 42 | sylib 207 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋) |
44 | 4, 43 | syl5ss 3579 |
. . . . . . 7
⊢ (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫
𝑋) |
45 | 44 | adantl 481 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋) |
46 | | ralss 3631 |
. . . . . 6
⊢
((𝒫 𝑎 ∩
Fin) ⊆ 𝒫 𝑋
→ (∀𝑐 ∈
(𝒫 𝑎 ∩
Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))) |
47 | 45, 46 | syl 17 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)))) |
48 | | bi2.04 375 |
. . . . . . 7
⊢ ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
49 | 48 | ralbii 2963 |
. . . . . 6
⊢
(∀𝑐 ∈
𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
50 | | elpwg 4116 |
. . . . . . . . 9
⊢ (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋)) |
51 | 50 | biimparc 503 |
. . . . . . . 8
⊢ ((𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋) |
52 | 51 | ad2antlr 759 |
. . . . . . 7
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋) |
53 | | eleq1 2676 |
. . . . . . . . 9
⊢ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin))) |
54 | 53 | imbi1d 330 |
. . . . . . . 8
⊢ (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
55 | 54 | ceqsralv 3207 |
. . . . . . 7
⊢ (𝑇 ∈ 𝒫 𝑋 → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
56 | 52, 55 | syl 17 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
57 | 49, 56 | syl5bb 271 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇 → 𝐾 ∈ 𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎))) |
58 | | vex 3176 |
. . . . . . . 8
⊢ 𝑎 ∈ V |
59 | 58 | elpw2 4755 |
. . . . . . 7
⊢ (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ⊆ 𝑎) |
60 | | simplrr 797 |
. . . . . . . . 9
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin) |
61 | 60 | biantrud 527 |
. . . . . . . 8
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin))) |
62 | | elin 3758 |
. . . . . . . 8
⊢ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎 ∧ 𝑇 ∈ Fin)) |
63 | 61, 62 | syl6bbr 277 |
. . . . . . 7
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin))) |
64 | 59, 63 | syl5rbbr 274 |
. . . . . 6
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ⊆ 𝑎)) |
65 | 64 | imbi1d 330 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎))) |
66 | 47, 57, 65 | 3bitrd 293 |
. . . 4
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇 → 𝐾 ∈ 𝑎) ↔ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎))) |
67 | 22, 41, 66 | 3bitrrd 294 |
. . 3
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎) ↔ ∪ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)) |
68 | 67 | rabbidva 3163 |
. 2
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∪
((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎}) |
69 | | simpll 786 |
. . 3
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → 𝑋 ∈ 𝑉) |
70 | | snelpwi 4839 |
. . . . . . 7
⊢ (𝐾 ∈ 𝑋 → {𝐾} ∈ 𝒫 𝑋) |
71 | 70 | ad2antlr 759 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋) |
72 | | 0elpw 4760 |
. . . . . 6
⊢ ∅
∈ 𝒫 𝑋 |
73 | | ifcl 4080 |
. . . . . 6
⊢ (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫
𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋) |
74 | 71, 72, 73 | sylancl 693 |
. . . . 5
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋) |
75 | 74 | adantr 480 |
. . . 4
⊢ ((((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋) |
76 | 75, 14 | fmptd 6292 |
. . 3
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋) |
77 | | isacs1i 16141 |
. . 3
⊢ ((𝑋 ∈ 𝑉 ∧ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∪
((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋)) |
78 | 69, 76, 77 | syl2anc 691 |
. 2
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ ∪
((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋)) |
79 | 68, 78 | eqeltrd 2688 |
1
⊢ (((𝑋 ∈ 𝑉 ∧ 𝐾 ∈ 𝑋) ∧ (𝑇 ⊆ 𝑋 ∧ 𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇 ⊆ 𝑎 → 𝐾 ∈ 𝑎)} ∈ (ACS‘𝑋)) |