Step | Hyp | Ref
| Expression |
1 | | fnresdm 6621 |
. . 3
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹) |
2 | 1 | adantr 482 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) = 𝐹) |
3 | | reseq2 5933 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐹 ↾ 𝑥) = (𝐹 ↾ ∅)) |
4 | 3 | eleq1d 2823 |
. . . . 5
⊢ (𝑥 = ∅ → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ ∅) ∈
Fin)) |
5 | 4 | imbi2d 341 |
. . . 4
⊢ (𝑥 = ∅ → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈
Fin))) |
6 | | reseq2 5933 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝑦)) |
7 | 6 | eleq1d 2823 |
. . . . 5
⊢ (𝑥 = 𝑦 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝑦) ∈ Fin)) |
8 | 7 | imbi2d 341 |
. . . 4
⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑦) ∈ Fin))) |
9 | | reseq2 5933 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ 𝑥) = (𝐹 ↾ (𝑦 ∪ {𝑧}))) |
10 | 9 | eleq1d 2823 |
. . . . 5
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin)) |
11 | 10 | imbi2d 341 |
. . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin))) |
12 | | reseq2 5933 |
. . . . . 6
⊢ (𝑥 = 𝐴 → (𝐹 ↾ 𝑥) = (𝐹 ↾ 𝐴)) |
13 | 12 | eleq1d 2823 |
. . . . 5
⊢ (𝑥 = 𝐴 → ((𝐹 ↾ 𝑥) ∈ Fin ↔ (𝐹 ↾ 𝐴) ∈ Fin)) |
14 | 13 | imbi2d 341 |
. . . 4
⊢ (𝑥 = 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝑥) ∈ Fin) ↔ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin))) |
15 | | res0 5942 |
. . . . . 6
⊢ (𝐹 ↾ ∅) =
∅ |
16 | | 0fin 9116 |
. . . . . 6
⊢ ∅
∈ Fin |
17 | 15, 16 | eqeltri 2834 |
. . . . 5
⊢ (𝐹 ↾ ∅) ∈
Fin |
18 | 17 | a1i 11 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ ∅) ∈
Fin) |
19 | | resundi 5952 |
. . . . . . . 8
⊢ (𝐹 ↾ (𝑦 ∪ {𝑧})) = ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) |
20 | | snfi 8989 |
. . . . . . . . . 10
⊢
{⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin |
21 | | fnfun 6603 |
. . . . . . . . . . . 12
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) |
22 | | funressn 7106 |
. . . . . . . . . . . 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 482 |
. . . . . . . . . 10
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩}) |
25 | | ssfi 9118 |
. . . . . . . . . 10
⊢
(({⟨𝑧, (𝐹‘𝑧)⟩} ∈ Fin ∧ (𝐹 ↾ {𝑧}) ⊆ {⟨𝑧, (𝐹‘𝑧)⟩}) → (𝐹 ↾ {𝑧}) ∈ Fin) |
26 | 20, 24, 25 | sylancr 588 |
. . . . . . . . 9
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ {𝑧}) ∈ Fin) |
27 | | unfi 9117 |
. . . . . . . . 9
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 ↾ {𝑧}) ∈ Fin) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin) |
28 | 26, 27 | sylan2 594 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → ((𝐹 ↾ 𝑦) ∪ (𝐹 ↾ {𝑧})) ∈ Fin) |
29 | 19, 28 | eqeltrid 2842 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝑦) ∈ Fin ∧ (𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin)) → (𝐹 ↾ (𝑦 ∪ {𝑧})) ∈ Fin) |
30 | 29 | expcom 415 |
. . . . . 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 9109 |
. . 3
⊢ (𝐴 ∈ Fin → ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin)) |
34 | 33 | anabsi7 670 |
. 2
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → (𝐹 ↾ 𝐴) ∈ Fin) |
35 | 2, 34 | eqeltrrd 2839 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) |