| Step | Hyp | Ref
| Expression |
| 1 | | fnresdm 6687 |
. . 3
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
| 2 | 1 | adantr 480 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹) |
| 3 | | reseq2 5992 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅)) |
| 4 | 3 | eleq1d 2826 |
. . . . 5
⊢ (𝑥 = ∅ → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈
Fin)) |
| 5 | 4 | imbi2d 340 |
. . . 4
⊢ (𝑥 = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈
Fin))) |
| 6 | | reseq2 5992 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦)) |
| 7 | 6 | eleq1d 2826 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin)) |
| 8 | 7 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin))) |
| 9 | | reseq2 5992 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧}))) |
| 10 | 9 | eleq1d 2826 |
. . . . 5
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)) |
| 11 | 10 | imbi2d 340 |
. . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))) |
| 12 | | reseq2 5992 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴)) |
| 13 | 12 | eleq1d 2826 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin)) |
| 14 | 13 | imbi2d 340 |
. . . 4
⊢ (𝑥 = 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin))) |
| 15 | | res0 6001 |
. . . . . 6
⊢ (𝐹 ↾ ∅) =
∅ |
| 16 | | 0fi 9082 |
. . . . . 6
⊢ ∅
∈ Fin |
| 17 | 15, 16 | eqeltri 2837 |
. . . . 5
⊢ (𝐹 ↾ ∅) ∈
Fin |
| 18 | 17 | a1i 11 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈
Fin) |
| 19 | | resundi 6011 |
. . . . . . . 8
⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) |
| 20 | | snfi 9083 |
. . . . . . . . . 10
⊢
{〈𝑧, (𝐹‘𝑧)〉} ∈ Fin |
| 21 | | fnfun 6668 |
. . . . . . . . . . . 12
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
| 22 | | funressn 7179 |
. . . . . . . . . . . 12
⊢ (Fun
𝐹 → (𝐹 ↾ {𝑧}) ⊆ {〈𝑧, (𝐹‘𝑧)〉}) |
| 23 | 21, 22 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ {𝑧}) ⊆ {〈𝑧, (𝐹‘𝑧)〉}) |
| 24 | 23 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {〈𝑧, (𝐹‘𝑧)〉}) |
| 25 | | ssfi 9213 |
. . . . . . . . . 10
⊢
(({〈𝑧, (𝐹‘𝑧)〉} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {〈𝑧, (𝐹‘𝑧)〉}) → (𝐹 ↾ {𝑧}) ∈ Fin) |
| 26 | 20, 24, 25 | sylancr 587 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin) |
| 27 | | unfi 9211 |
. . . . . . . . 9
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin) |
| 28 | 26, 27 | sylan2 593 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin) |
| 29 | 19, 28 | eqeltrid 2845 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin) |
| 30 | 29 | expcom 413 |
. . . . . 6
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → ((𝐹 ↾ 𝑦) ∈ Fin → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)) |
| 31 | 30 | a2i 14 |
. . . . 5
⊢ (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)) |
| 32 | 31 | a1i 11 |
. . . 4
⊢ (𝑦 ∈ Fin → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin) → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))) |
| 33 | 5, 8, 11, 14, 18, 32 | findcard2 9204 |
. . 3
⊢ (𝐴 ∈ Fin → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)) |
| 34 | 33 | anabsi7 671 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin) |
| 35 | 2, 34 | eqeltrrd 2842 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) |