| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-card 9980 | . . . 4
⊢ card =
(𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥}) | 
| 2 | 1 | funmpt2 6604 | . . 3
⊢ Fun
card | 
| 3 |  | rabab 3511 | . . . 4
⊢ {𝑥 ∈ V ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈ V} = {𝑥 ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V} | 
| 4 | 1 | dmmpt 6259 | . . . 4
⊢ dom card
= {𝑥 ∈ V ∣ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V} | 
| 5 |  | intexrab 5346 | . . . . 5
⊢
(∃𝑦 ∈ On
𝑦 ≈ 𝑥 ↔ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥} ∈
V) | 
| 6 | 5 | abbii 2808 | . . . 4
⊢ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} = {𝑥 ∣ ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑥} ∈ V} | 
| 7 | 3, 4, 6 | 3eqtr4i 2774 | . . 3
⊢ dom card
= {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} | 
| 8 |  | df-fn 6563 | . . 3
⊢ (card Fn
{𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ (Fun card ∧ dom card = {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥})) | 
| 9 | 2, 7, 8 | mpbir2an 711 | . 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 3483 | . . . . . . . . 9
⊢ 𝑤 ∈ V | 
| 12 | 10, 11 | eqeltrrdi 2849 | . . . . . . . 8
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ V) | 
| 13 |  | intex 5343 | . . . . . . . 8
⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧} ∈
V) | 
| 14 | 12, 13 | sylibr 234 | . . . . . . 7
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) | 
| 15 |  | rabn0 4388 | . . . . . . 7
⊢ ({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅ ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧) | 
| 16 | 14, 15 | sylib 218 | . . . . . 6
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∃𝑦 ∈ On 𝑦 ≈ 𝑧) | 
| 17 |  | vex 3483 | . . . . . . 7
⊢ 𝑧 ∈ V | 
| 18 |  | breq2 5146 | . . . . . . . 8
⊢ (𝑥 = 𝑧 → (𝑦 ≈ 𝑥 ↔ 𝑦 ≈ 𝑧)) | 
| 19 | 18 | rexbidv 3178 | . . . . . . 7
⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ On 𝑦 ≈ 𝑥 ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧)) | 
| 20 | 17, 19 | elab 3678 | . . . . . 6
⊢ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝑧) | 
| 21 | 16, 20 | sylibr 234 | . . . . 5
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → 𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥}) | 
| 22 |  | ssrab2 4079 | . . . . . . 7
⊢ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On | 
| 23 |  | oninton 7816 | . . . . . . 7
⊢ (({𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ⊆ On ∧ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ≠ ∅) → ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧} ∈
On) | 
| 24 | 22, 14, 23 | sylancr 587 | . . . . . 6
⊢ ((𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧}) → ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧} ∈ On) | 
| 25 | 10, 24 | eqeltrd 2840 | . . . . 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 5554 | . . 3
⊢
{〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} ⊆ {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)} | 
| 28 |  | df-card 9980 | . . . 4
⊢ card =
(𝑧 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧}) | 
| 29 |  | df-mpt 5225 | . . . 4
⊢ (𝑧 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑧}) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩ {𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} | 
| 30 | 28, 29 | eqtri 2764 | . . 3
⊢ card =
{〈𝑧, 𝑤〉 ∣ (𝑧 ∈ V ∧ 𝑤 = ∩
{𝑦 ∈ On ∣ 𝑦 ≈ 𝑧})} | 
| 31 |  | df-xp 5690 | . . 3
⊢ ({𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) = {〈𝑧, 𝑤〉 ∣ (𝑧 ∈ {𝑥 ∣ ∃𝑦 ∈ On 𝑦 ≈ 𝑥} ∧ 𝑤 ∈ On)} | 
| 32 | 27, 30, 31 | 3sstr4i 4034 | . 2
⊢ card
⊆ ({𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥} × On) | 
| 33 |  | dff2 7118 | . 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 711 | 1
⊢
card:{𝑥 ∣
∃𝑦 ∈ On 𝑦 ≈ 𝑥}⟶On |