| Step | Hyp | Ref
| Expression |
| 1 | | cfval 10287 |
. . 3
⊢ (𝐴 ∈ On →
(cf‘𝐴) = ∩ {𝑥
∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
| 2 | | df-sn 4627 |
. . . . . 6
⊢
{(card‘𝐴)} =
{𝑥 ∣ 𝑥 = (card‘𝐴)} |
| 3 | | ssid 4006 |
. . . . . . . . 9
⊢ 𝐴 ⊆ 𝐴 |
| 4 | | ssid 4006 |
. . . . . . . . . . 11
⊢ 𝑧 ⊆ 𝑧 |
| 5 | | sseq2 4010 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑧 → (𝑧 ⊆ 𝑤 ↔ 𝑧 ⊆ 𝑧)) |
| 6 | 5 | rspcev 3622 |
. . . . . . . . . . 11
⊢ ((𝑧 ∈ 𝐴 ∧ 𝑧 ⊆ 𝑧) → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
| 7 | 4, 6 | mpan2 691 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝐴 → ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
| 8 | 7 | rgen 3063 |
. . . . . . . . 9
⊢
∀𝑧 ∈
𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤 |
| 9 | 3, 8 | pm3.2i 470 |
. . . . . . . 8
⊢ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤) |
| 10 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (card‘𝑦) = (card‘𝐴)) |
| 11 | 10 | eqeq2d 2748 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (𝑥 = (card‘𝑦) ↔ 𝑥 = (card‘𝐴))) |
| 12 | | sseq1 4009 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴)) |
| 13 | | rexeq 3322 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝐴 → (∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) |
| 14 | 13 | ralbidv 3178 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)) |
| 15 | 12, 14 | anbi12d 632 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → ((𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤) ↔ (𝐴 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤))) |
| 16 | 11, 15 | anbi12d 632 |
. . . . . . . . 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 3597 |
. . . . . . . 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 697 |
. . . . . . 7
⊢ (𝐴 ∈ On → (𝑥 = (card‘𝐴) → ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)))) |
| 19 | 18 | ss2abdv 4066 |
. . . . . 6
⊢ (𝐴 ∈ On → {𝑥 ∣ 𝑥 = (card‘𝐴)} ⊆ {𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
| 20 | 2, 19 | eqsstrid 4022 |
. . . . 5
⊢ (𝐴 ∈ On →
{(card‘𝐴)} ⊆
{𝑥 ∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))}) |
| 21 | | intss 4969 |
. . . . 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 6919 |
. . . . 5
⊢
(card‘𝐴)
∈ V |
| 24 | 23 | intsn 4984 |
. . . 4
⊢ ∩ {(card‘𝐴)} = (card‘𝐴) |
| 25 | 22, 24 | sseqtrdi 4024 |
. . 3
⊢ (𝐴 ∈ On → ∩ {𝑥
∣ ∃𝑦(𝑥 = (card‘𝑦) ∧ (𝑦 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤))} ⊆ (card‘𝐴)) |
| 26 | 1, 25 | eqsstrd 4018 |
. 2
⊢ (𝐴 ∈ On →
(cf‘𝐴) ⊆
(card‘𝐴)) |
| 27 | | cff 10288 |
. . . . . 6
⊢
cf:On⟶On |
| 28 | 27 | fdmi 6747 |
. . . . 5
⊢ dom cf =
On |
| 29 | 28 | eleq2i 2833 |
. . . 4
⊢ (𝐴 ∈ dom cf ↔ 𝐴 ∈ On) |
| 30 | | ndmfv 6941 |
. . . 4
⊢ (¬
𝐴 ∈ dom cf →
(cf‘𝐴) =
∅) |
| 31 | 29, 30 | sylnbir 331 |
. . 3
⊢ (¬
𝐴 ∈ On →
(cf‘𝐴) =
∅) |
| 32 | | 0ss 4400 |
. . 3
⊢ ∅
⊆ (card‘𝐴) |
| 33 | 31, 32 | eqsstrdi 4028 |
. 2
⊢ (¬
𝐴 ∈ On →
(cf‘𝐴) ⊆
(card‘𝐴)) |
| 34 | 26, 33 | pm2.61i 182 |
1
⊢
(cf‘𝐴) ⊆
(card‘𝐴) |