Step | Hyp | Ref
| Expression |
1 | | cfval 9069 |
. . 3
⊢ (𝐴 ∈ On →
(cf‘𝐴) = ∩ {𝑥
∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
2 | | df-sn 4178 |
. . . . . 6
⊢
{(card‘𝐴)} =
{𝑥 ∣ 𝑥 = (card‘𝐴)} |
3 | | ssid 3624 |
. . . . . . . . 9
⊢ 𝐴 ⊆ 𝐴 |
4 | | ssid 3624 |
. . . . . . . . . . 11
⊢ 𝑧 ⊆ 𝑧 |
5 | | sseq2 3627 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧)) |
6 | 5 | rspcev 3309 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
7 | 4, 6 | mpan2 707 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
8 | 7 | rgen 2922 |
. . . . . . . . 9
⊢
∀𝑧 ∈
𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤 |
9 | 3, 8 | pm3.2i 471 |
. . . . . . . 8
⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
10 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴)) |
11 | 10 | eqeq2d 2632 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴))) |
12 | | sseq1 3626 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
13 | | rexeq 3139 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) |
14 | 13 | ralbidv 2986 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) |
15 | 12, 14 | anbi12d 747 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))) |
16 | 11, 15 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → ((𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)) ↔ (𝑥 = (card‘𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)))) |
17 | 16 | spcegv 3294 |
. . . . . . . 8
⊢ (𝐴 ∈ On → ((𝑥 = (card‘𝐴) ∧ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))) |
18 | 9, 17 | mpan2i 713 |
. . . . . . 7
⊢ (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))) |
19 | 18 | ss2abdv 3675 |
. . . . . 6
⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
20 | 2, 19 | syl5eqss 3649 |
. . . . 5
⊢ (𝐴 ∈ On →
{(card‘𝐴)} ⊆
{𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
21 | | intss 4498 |
. . . . 5
⊢
({(card‘𝐴)}
⊆ {𝑥 ∣
∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} → ∩
{𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩
{(card‘𝐴)}) |
22 | 20, 21 | syl 17 |
. . . 4
⊢ (𝐴 ∈ On → ∩ {𝑥
∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ ∩
{(card‘𝐴)}) |
23 | | fvex 6201 |
. . . . 5
⊢
(card‘𝐴)
∈ V |
24 | 23 | intsn 4513 |
. . . 4
⊢ ∩ {(card‘𝐴)} = (card‘𝐴) |
25 | 22, 24 | syl6sseq 3651 |
. . 3
⊢ (𝐴 ∈ On → ∩ {𝑥
∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴)) |
26 | 1, 25 | eqsstrd 3639 |
. 2
⊢ (𝐴 ∈ On →
(cf‘𝐴) ⊆
(card‘𝐴)) |
27 | | cff 9070 |
. . . . . 6
⊢
cf:On⟶On |
28 | 27 | fdmi 6052 |
. . . . 5
⊢ dom cf =
On |
29 | 28 | eleq2i 2693 |
. . . 4
⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) |
30 | | ndmfv 6218 |
. . . 4
⊢ (¬
𝐴 ∈ dom cf →
(cf‘𝐴) =
∅) |
31 | 29, 30 | sylnbir 321 |
. . 3
⊢ (¬
𝐴 ∈ On →
(cf‘𝐴) =
∅) |
32 | | 0ss 3972 |
. . 3
⊢ ∅
⊆ (card‘𝐴) |
33 | 31, 32 | syl6eqss 3655 |
. 2
⊢ (¬
𝐴 ∈ On →
(cf‘𝐴) ⊆
(card‘𝐴)) |
34 | 26, 33 | pm2.61i 176 |
1
⊢
(cf‘𝐴) ⊆
(card‘𝐴) |