| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | harval2 10038 | . 2
⊢ (𝐴 ∈ dom card →
(har‘𝐴) = ∩ {𝑦
∈ On ∣ 𝐴 ≺
𝑦}) | 
| 2 |  | vex 3483 | . . . . . 6
⊢ 𝑥 ∈ V | 
| 3 | 2 | a1i 11 | . . . . 5
⊢ (𝐴 ∈ dom card → 𝑥 ∈ V) | 
| 4 |  | elrncard 43555 | . . . . . . . . 9
⊢ (𝑥 ∈ ran card ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 ¬ 𝑦 ≈ 𝑥)) | 
| 5 | 4 | simplbi 497 | . . . . . . . 8
⊢ (𝑥 ∈ ran card → 𝑥 ∈ On) | 
| 6 | 5 | anim1i 615 | . . . . . . 7
⊢ ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥)) | 
| 7 |  | eleq1 2828 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → (𝑦 ∈ On ↔ 𝑥 ∈ On)) | 
| 8 |  | breq2 5146 | . . . . . . . 8
⊢ (𝑦 = 𝑥 → (𝐴 ≺ 𝑦 ↔ 𝐴 ≺ 𝑥)) | 
| 9 | 7, 8 | anbi12d 632 | . . . . . . 7
⊢ (𝑦 = 𝑥 → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) ↔ (𝑥 ∈ On ∧ 𝐴 ≺ 𝑥))) | 
| 10 | 6, 9 | imbitrrid 246 | . . . . . 6
⊢ (𝑦 = 𝑥 → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))) | 
| 11 | 10 | adantl 481 | . . . . 5
⊢ ((𝐴 ∈ dom card ∧ 𝑦 = 𝑥) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) → (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦))) | 
| 12 |  | ssidd 4006 | . . . . 5
⊢ (𝐴 ∈ dom card → 𝑥 ⊆ 𝑥) | 
| 13 | 3, 11, 12 | intabssd 43537 | . . . 4
⊢ (𝐴 ∈ dom card → ∩ {𝑦
∣ (𝑦 ∈ On ∧
𝐴 ≺ 𝑦)} ⊆ ∩ {𝑥
∣ (𝑥 ∈ ran card
∧ 𝐴 ≺ 𝑥)}) | 
| 14 |  | vex 3483 | . . . . . . 7
⊢ 𝑦 ∈ V | 
| 15 | 14 | inex1 5316 | . . . . . 6
⊢ (𝑦 ∩ (card‘𝑦)) ∈ V | 
| 16 | 15 | a1i 11 | . . . . 5
⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ∈ V) | 
| 17 |  | oncardid 9997 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ On →
(card‘𝑦) ≈
𝑦) | 
| 18 | 17 | ensymd 9046 | . . . . . . . . . . 11
⊢ (𝑦 ∈ On → 𝑦 ≈ (card‘𝑦)) | 
| 19 |  | sdomentr 9152 | . . . . . . . . . . . 12
⊢ ((𝐴 ≺ 𝑦 ∧ 𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦)) | 
| 20 | 19 | a1i 11 | . . . . . . . . . . 11
⊢ (𝑦 ∈ On → ((𝐴 ≺ 𝑦 ∧ 𝑦 ≈ (card‘𝑦)) → 𝐴 ≺ (card‘𝑦))) | 
| 21 | 18, 20 | mpan2d 694 | . . . . . . . . . 10
⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → 𝐴 ≺ (card‘𝑦))) | 
| 22 |  | df-card 9980 | . . . . . . . . . . . 12
⊢ card =
(𝑥 ∈ V ↦ ∩ {𝑦
∈ On ∣ 𝑦 ≈
𝑥}) | 
| 23 | 22 | funmpt2 6604 | . . . . . . . . . . 11
⊢ Fun
card | 
| 24 |  | onenon 9990 | . . . . . . . . . . 11
⊢ (𝑦 ∈ On → 𝑦 ∈ dom
card) | 
| 25 |  | fvelrn 7095 | . . . . . . . . . . 11
⊢ ((Fun
card ∧ 𝑦 ∈ dom
card) → (card‘𝑦)
∈ ran card) | 
| 26 | 23, 24, 25 | sylancr 587 | . . . . . . . . . 10
⊢ (𝑦 ∈ On →
(card‘𝑦) ∈ ran
card) | 
| 27 | 21, 26 | jctild 525 | . . . . . . . . 9
⊢ (𝑦 ∈ On → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦)))) | 
| 28 | 27 | adantl 481 | . . . . . . . 8
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦)))) | 
| 29 |  | simpl 482 | . . . . . . . . . 10
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (𝑦 ∩ (card‘𝑦))) | 
| 30 |  | cardonle 9998 | . . . . . . . . . . . 12
⊢ (𝑦 ∈ On →
(card‘𝑦) ⊆
𝑦) | 
| 31 | 30 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (card‘𝑦) ⊆ 𝑦) | 
| 32 |  | sseqin2 4222 | . . . . . . . . . . 11
⊢
((card‘𝑦)
⊆ 𝑦 ↔ (𝑦 ∩ (card‘𝑦)) = (card‘𝑦)) | 
| 33 | 31, 32 | sylib 218 | . . . . . . . . . 10
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝑦 ∩ (card‘𝑦)) = (card‘𝑦)) | 
| 34 | 29, 33 | eqtrd 2776 | . . . . . . . . 9
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → 𝑥 = (card‘𝑦)) | 
| 35 |  | eleq1 2828 | . . . . . . . . . 10
⊢ (𝑥 = (card‘𝑦) → (𝑥 ∈ ran card ↔ (card‘𝑦) ∈ ran
card)) | 
| 36 |  | breq2 5146 | . . . . . . . . . 10
⊢ (𝑥 = (card‘𝑦) → (𝐴 ≺ 𝑥 ↔ 𝐴 ≺ (card‘𝑦))) | 
| 37 | 35, 36 | anbi12d 632 | . . . . . . . . 9
⊢ (𝑥 = (card‘𝑦) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦)))) | 
| 38 | 34, 37 | syl 17 | . . . . . . . 8
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → ((𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥) ↔ ((card‘𝑦) ∈ ran card ∧ 𝐴 ≺ (card‘𝑦)))) | 
| 39 | 28, 38 | sylibrd 259 | . . . . . . 7
⊢ ((𝑥 = (𝑦 ∩ (card‘𝑦)) ∧ 𝑦 ∈ On) → (𝐴 ≺ 𝑦 → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥))) | 
| 40 | 39 | expimpd 453 | . . . . . 6
⊢ (𝑥 = (𝑦 ∩ (card‘𝑦)) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥))) | 
| 41 | 40 | adantl 481 | . . . . 5
⊢ ((𝐴 ∈ dom card ∧ 𝑥 = (𝑦 ∩ (card‘𝑦))) → ((𝑦 ∈ On ∧ 𝐴 ≺ 𝑦) → (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥))) | 
| 42 |  | inss1 4236 | . . . . . 6
⊢ (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦 | 
| 43 | 42 | a1i 11 | . . . . 5
⊢ (𝐴 ∈ dom card → (𝑦 ∩ (card‘𝑦)) ⊆ 𝑦) | 
| 44 | 16, 41, 43 | intabssd 43537 | . . . 4
⊢ (𝐴 ∈ dom card → ∩ {𝑥
∣ (𝑥 ∈ ran card
∧ 𝐴 ≺ 𝑥)} ⊆ ∩ {𝑦
∣ (𝑦 ∈ On ∧
𝐴 ≺ 𝑦)}) | 
| 45 | 13, 44 | eqssd 4000 | . . 3
⊢ (𝐴 ∈ dom card → ∩ {𝑦
∣ (𝑦 ∈ On ∧
𝐴 ≺ 𝑦)} = ∩ {𝑥
∣ (𝑥 ∈ ran card
∧ 𝐴 ≺ 𝑥)}) | 
| 46 |  | df-rab 3436 | . . . 4
⊢ {𝑦 ∈ On ∣ 𝐴 ≺ 𝑦} = {𝑦 ∣ (𝑦 ∈ On ∧ 𝐴 ≺ 𝑦)} | 
| 47 | 46 | inteqi 4949 | . . 3
⊢ ∩ {𝑦
∈ On ∣ 𝐴 ≺
𝑦} = ∩ {𝑦
∣ (𝑦 ∈ On ∧
𝐴 ≺ 𝑦)} | 
| 48 |  | df-rab 3436 | . . . 4
⊢ {𝑥 ∈ ran card ∣ 𝐴 ≺ 𝑥} = {𝑥 ∣ (𝑥 ∈ ran card ∧ 𝐴 ≺ 𝑥)} | 
| 49 | 48 | inteqi 4949 | . . 3
⊢ ∩ {𝑥
∈ ran card ∣ 𝐴
≺ 𝑥} = ∩ {𝑥
∣ (𝑥 ∈ ran card
∧ 𝐴 ≺ 𝑥)} | 
| 50 | 45, 47, 49 | 3eqtr4g 2801 | . 2
⊢ (𝐴 ∈ dom card → ∩ {𝑦
∈ On ∣ 𝐴 ≺
𝑦} = ∩ {𝑥
∈ ran card ∣ 𝐴
≺ 𝑥}) | 
| 51 | 1, 50 | eqtrd 2776 | 1
⊢ (𝐴 ∈ dom card →
(har‘𝐴) = ∩ {𝑥
∈ ran card ∣ 𝐴
≺ 𝑥}) |