Step | Hyp | Ref
| Expression |
1 | | pwundif 4565 |
. 2
⊢ 𝒫
(𝑏 ∪ {𝑥}) = ((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∪ 𝒫 𝑏) |
2 | | vex 3497 |
. . . . . . . . 9
⊢ 𝑐 ∈ V |
3 | | snex 5332 |
. . . . . . . . 9
⊢ {𝑥} ∈ V |
4 | 2, 3 | unex 7469 |
. . . . . . . 8
⊢ (𝑐 ∪ {𝑥}) ∈ V |
5 | | pwfilem.1 |
. . . . . . . 8
⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥})) |
6 | 4, 5 | fnmpti 6491 |
. . . . . . 7
⊢ 𝐹 Fn 𝒫 𝑏 |
7 | | dffn4 6596 |
. . . . . . 7
⊢ (𝐹 Fn 𝒫 𝑏 ↔ 𝐹:𝒫 𝑏–onto→ran 𝐹) |
8 | 6, 7 | mpbi 232 |
. . . . . 6
⊢ 𝐹:𝒫 𝑏–onto→ran 𝐹 |
9 | | fodomfi 8797 |
. . . . . 6
⊢
((𝒫 𝑏 ∈
Fin ∧ 𝐹:𝒫 𝑏–onto→ran 𝐹) → ran 𝐹 ≼ 𝒫 𝑏) |
10 | 8, 9 | mpan2 689 |
. . . . 5
⊢
(𝒫 𝑏 ∈
Fin → ran 𝐹 ≼
𝒫 𝑏) |
11 | | domfi 8739 |
. . . . 5
⊢
((𝒫 𝑏 ∈
Fin ∧ ran 𝐹 ≼
𝒫 𝑏) → ran
𝐹 ∈
Fin) |
12 | 10, 11 | mpdan 685 |
. . . 4
⊢
(𝒫 𝑏 ∈
Fin → ran 𝐹 ∈
Fin) |
13 | | eldifi 4103 |
. . . . . . . . 9
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥})) |
14 | 3 | elpwun 7491 |
. . . . . . . . 9
⊢ (𝑑 ∈ 𝒫 (𝑏 ∪ {𝑥}) ↔ (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏) |
15 | 13, 14 | sylib 220 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏) |
16 | | undif1 4424 |
. . . . . . . . 9
⊢ ((𝑑 ∖ {𝑥}) ∪ {𝑥}) = (𝑑 ∪ {𝑥}) |
17 | | elpwunsn 4621 |
. . . . . . . . . . 11
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑥 ∈ 𝑑) |
18 | 17 | snssd 4742 |
. . . . . . . . . 10
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → {𝑥} ⊆ 𝑑) |
19 | | ssequn2 4159 |
. . . . . . . . . 10
⊢ ({𝑥} ⊆ 𝑑 ↔ (𝑑 ∪ {𝑥}) = 𝑑) |
20 | 18, 19 | sylib 220 |
. . . . . . . . 9
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → (𝑑 ∪ {𝑥}) = 𝑑) |
21 | 16, 20 | syl5req 2869 |
. . . . . . . 8
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) |
22 | | uneq1 4132 |
. . . . . . . . 9
⊢ (𝑐 = (𝑑 ∖ {𝑥}) → (𝑐 ∪ {𝑥}) = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) |
23 | 22 | rspceeqv 3638 |
. . . . . . . 8
⊢ (((𝑑 ∖ {𝑥}) ∈ 𝒫 𝑏 ∧ 𝑑 = ((𝑑 ∖ {𝑥}) ∪ {𝑥})) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥})) |
24 | 15, 21, 23 | syl2anc 586 |
. . . . . . 7
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥})) |
25 | 5, 4 | elrnmpti 5832 |
. . . . . . 7
⊢ (𝑑 ∈ ran 𝐹 ↔ ∃𝑐 ∈ 𝒫 𝑏𝑑 = (𝑐 ∪ {𝑥})) |
26 | 24, 25 | sylibr 236 |
. . . . . 6
⊢ (𝑑 ∈ (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) → 𝑑 ∈ ran 𝐹) |
27 | 26 | ssriv 3971 |
. . . . 5
⊢
(𝒫 (𝑏 ∪
{𝑥}) ∖ 𝒫
𝑏) ⊆ ran 𝐹 |
28 | | ssdomg 8555 |
. . . . 5
⊢ (ran
𝐹 ∈ Fin →
((𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ⊆ ran 𝐹 → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹)) |
29 | 12, 27, 28 | mpisyl 21 |
. . . 4
⊢
(𝒫 𝑏 ∈
Fin → (𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ≼ ran
𝐹) |
30 | | domfi 8739 |
. . . 4
⊢ ((ran
𝐹 ∈ Fin ∧
(𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ≼ ran 𝐹) → (𝒫 (𝑏 ∪ {𝑥}) ∖ 𝒫 𝑏) ∈ Fin) |
31 | 12, 29, 30 | syl2anc 586 |
. . 3
⊢
(𝒫 𝑏 ∈
Fin → (𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∈
Fin) |
32 | | unfi 8785 |
. . 3
⊢
(((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∈ Fin
∧ 𝒫 𝑏 ∈
Fin) → ((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∪
𝒫 𝑏) ∈
Fin) |
33 | 31, 32 | mpancom 686 |
. 2
⊢
(𝒫 𝑏 ∈
Fin → ((𝒫 (𝑏
∪ {𝑥}) ∖
𝒫 𝑏) ∪
𝒫 𝑏) ∈
Fin) |
34 | 1, 33 | eqeltrid 2917 |
1
⊢
(𝒫 𝑏 ∈
Fin → 𝒫 (𝑏
∪ {𝑥}) ∈
Fin) |