Step | Hyp | Ref
| Expression |
1 | | df-card 9725 |
. . . 4
⊢ card =
(𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥}) |
2 | 1 | funmpt2 6490 |
. . 3
⊢ Fun
card |
3 | | rabab 3462 |
. . . 4
⊢ {𝑥 ∈ V ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V} |
4 | 1 | dmmpt 6147 |
. . . 4
⊢ dom card
= {𝑥 ∈ V ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V} |
5 | | intexrab 5267 |
. . . . 5
⊢
(∃𝑦 ∈ On
𝑦 ≈ 𝑥 ↔ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V) |
6 | 5 | abbii 2803 |
. . . 4
⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} |
7 | 3, 4, 6 | 3eqtr4i 2771 |
. . 3
⊢ dom card
= {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} |
8 | | df-fn 6450 |
. . 3
⊢ (card Fn
{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥})) |
9 | 2, 7, 8 | mpbir2an 707 |
. 2
⊢ card Fn
{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} |
10 | | simpr 484 |
. . . . . . . . 9
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) |
11 | | vex 3438 |
. . . . . . . . 9
⊢ 𝑤 ∈ V |
12 | 10, 11 | eqeltrrdi 2843 |
. . . . . . . 8
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V) |
13 | | intex 5264 |
. . . . . . . 8
⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧} ∈
V) |
14 | 12, 13 | sylibr 233 |
. . . . . . 7
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) |
15 | | rabn0 4322 |
. . . . . . 7
⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧) |
16 | 14, 15 | sylib 217 |
. . . . . 6
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∃𝑦 ∈ On 𝑦 ≈ 𝑧) |
17 | | vex 3438 |
. . . . . . 7
⊢ 𝑧 ∈ V |
18 | | breq2 5081 |
. . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧)) |
19 | 18 | rexbidv 3169 |
. . . . . . 7
⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)) |
20 | 17, 19 | elab 3611 |
. . . . . 6
⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧) |
21 | 16, 20 | sylibr 233 |
. . . . 5
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}) |
22 | | ssrab2 4016 |
. . . . . . 7
⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On |
23 | | oninton 7665 |
. . . . . . 7
⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) → ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧} ∈
On) |
24 | 22, 14, 23 | sylancr 586 |
. . . . . 6
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On) |
25 | 10, 24 | eqeltrd 2834 |
. . . . 5
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑤 ∈ On) |
26 | 21, 25 | jca 511 |
. . . 4
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)) |
27 | 26 | ssopab2i 5466 |
. . 3
⊢
{〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)} |
28 | | df-card 9725 |
. . . 4
⊢ card =
(𝑧 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧}) |
29 | | df-mpt 5161 |
. . . 4
⊢ (𝑧 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧}) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} |
30 | 28, 29 | eqtri 2761 |
. . 3
⊢ card =
{〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} |
31 | | df-xp 5597 |
. . 3
⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)} |
32 | 27, 30, 31 | 3sstr4i 3966 |
. 2
⊢ card
⊆ ({𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) |
33 | | dff2 6995 |
. 2
⊢
(card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On ↔ (card Fn {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ card ⊆ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On))) |
34 | 9, 32, 33 | mpbir2an 707 |
1
⊢
card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On |