Step | Hyp | Ref
| Expression |
1 | | pwundif 4559 |
. 2
⊢ 𝒫
(𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) |
2 | | pwfilem.1 |
. . . . . 6
⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥})) |
3 | 2 | funmpt2 6473 |
. . . . 5
⊢ Fun 𝐹 |
4 | | vex 3436 |
. . . . . . . . . 10
⊢ 𝑐 ∈ V |
5 | | snex 5354 |
. . . . . . . . . 10
⊢ {𝑥} ∈ V |
6 | 4, 5 | unex 7596 |
. . . . . . . . 9
⊢ (𝑐 ∪ {𝑥}) ∈ V |
7 | 6, 2 | dmmpti 6577 |
. . . . . . . 8
⊢ dom 𝐹 = 𝒫 𝑏 |
8 | 7 | imaeq2i 5967 |
. . . . . . 7
⊢ (𝐹 “ dom 𝐹) = (𝐹 “ 𝒫 𝑏) |
9 | | imadmrn 5979 |
. . . . . . 7
⊢ (𝐹 “ dom 𝐹) = ran 𝐹 |
10 | 8, 9 | eqtr3i 2768 |
. . . . . 6
⊢ (𝐹 “ 𝒫 𝑏) = ran 𝐹 |
11 | | imafi 8958 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝒫 𝑏 ∈ Fin) → (𝐹 “ 𝒫 𝑏) ∈ Fin) |
12 | 10, 11 | eqeltrrid 2844 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝒫 𝑏 ∈ Fin) → ran 𝐹 ∈ Fin) |
13 | 3, 12 | mpan 687 |
. . . 4
⊢
(𝒫 𝑏 ∈
Fin → ran 𝐹 ∈
Fin) |
14 | | eldifi 4061 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥})) |
15 | 5 | elpwun 7619 |
. . . . . . . 8
⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏) |
16 | 14, 15 | sylib 217 |
. . . . . . 7
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏) |
17 | | undif1 4409 |
. . . . . . . 8
⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥}) |
18 | | elpwunsn 4619 |
. . . . . . . . . 10
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑) |
19 | 18 | snssd 4742 |
. . . . . . . . 9
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑) |
20 | | ssequn2 4117 |
. . . . . . . . 9
⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑) |
21 | 19, 20 | sylib 217 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑) |
22 | 17, 21 | eqtr2id 2791 |
. . . . . . 7
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) |
23 | | uneq1 4090 |
. . . . . . . 8
⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) |
24 | 23 | rspceeqv 3575 |
. . . . . . 7
⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥})) |
25 | 16, 22, 24 | syl2anc 584 |
. . . . . 6
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥})) |
26 | 2, 25, 14 | elrnmptd 5870 |
. . . . 5
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹) |
27 | 26 | ssriv 3925 |
. . . 4
⊢
(𝒫 (𝑏 ∪
{𝑥}) ∖ 𝒫
𝑏) ⊆ ran 𝐹 |
28 | | ssfi 8956 |
. . . 4
⊢ ((ran
𝐹 ∈ Fin ∧
(𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin) |
29 | 13, 27, 28 | sylancl 586 |
. . 3
⊢
(𝒫 𝑏 ∈
Fin → (𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∈
Fin) |
30 | | unfi 8955 |
. . 3
⊢
(((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∈ Fin
∧ 𝒫 𝑏 ∈
Fin) → ((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∪
𝒫 𝑏) ∈
Fin) |
31 | 29, 30 | mpancom 685 |
. 2
⊢
(𝒫 𝑏 ∈
Fin → ((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∪
𝒫 𝑏) ∈
Fin) |
32 | 1, 31 | eqeltrid 2843 |
1
⊢
(𝒫 𝑏 ∈
Fin → 𝒫 (𝑏
∪ {𝑥}) ∈
Fin) |