Step | Hyp | Ref
| Expression |
1 | | sneq 4568 |
. . . . . 6
⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) |
2 | | reseq2 5864 |
. . . . . . . 8
⊢ ({𝑥} = {𝐵} → (𝐹 ↾ {𝑥}) = (𝐹 ↾ {𝐵})) |
3 | 2 | feq1d 6552 |
. . . . . . 7
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝑥}⟶𝐶)) |
4 | | feq2 6549 |
. . . . . . 7
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝐵}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) |
5 | 3, 4 | bitrd 282 |
. . . . . 6
⊢ ({𝑥} = {𝐵} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) |
6 | 1, 5 | syl 17 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹 ↾ {𝐵}):{𝐵}⟶𝐶)) |
7 | | fveq2 6739 |
. . . . . 6
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
8 | 7 | eleq1d 2824 |
. . . . 5
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)) |
9 | 6, 8 | bibi12d 349 |
. . . 4
⊢ (𝑥 = 𝐵 → (((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) ↔ ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))) |
10 | 9 | imbi2d 344 |
. . 3
⊢ (𝑥 = 𝐵 → ((𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) ↔ (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)))) |
11 | | fnressn 6995 |
. . . . 5
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) |
12 | | vsnid 4595 |
. . . . . . . . . 10
⊢ 𝑥 ∈ {𝑥} |
13 | | fvres 6758 |
. . . . . . . . . 10
⊢ (𝑥 ∈ {𝑥} → ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥)) |
14 | 12, 13 | ax-mp 5 |
. . . . . . . . 9
⊢ ((𝐹 ↾ {𝑥})‘𝑥) = (𝐹‘𝑥) |
15 | 14 | opeq2i 4805 |
. . . . . . . 8
⊢
〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉 |
16 | 15 | sneqi 4569 |
. . . . . . 7
⊢
{〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} = {〈𝑥, (𝐹‘𝑥)〉} |
17 | 16 | eqeq2i 2752 |
. . . . . 6
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} ↔ (𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉}) |
18 | | vex 3427 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
19 | 18 | fsn2 6973 |
. . . . . . 7
⊢ ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉})) |
20 | | iba 531 |
. . . . . . . 8
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉}))) |
21 | 14 | eleq1i 2830 |
. . . . . . . 8
⊢ (((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶) |
22 | 20, 21 | bitr3di 289 |
. . . . . . 7
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → ((((𝐹 ↾ {𝑥})‘𝑥) ∈ 𝐶 ∧ (𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉}) ↔ (𝐹‘𝑥) ∈ 𝐶)) |
23 | 19, 22 | syl5bb 286 |
. . . . . 6
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, ((𝐹 ↾ {𝑥})‘𝑥)〉} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) |
24 | 17, 23 | sylbir 238 |
. . . . 5
⊢ ((𝐹 ↾ {𝑥}) = {〈𝑥, (𝐹‘𝑥)〉} → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) |
25 | 11, 24 | syl 17 |
. . . 4
⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶)) |
26 | 25 | expcom 417 |
. . 3
⊢ (𝑥 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝑥}):{𝑥}⟶𝐶 ↔ (𝐹‘𝑥) ∈ 𝐶))) |
27 | 10, 26 | vtoclga 3504 |
. 2
⊢ (𝐵 ∈ 𝐴 → (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶))) |
28 | 27 | impcom 411 |
1
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹 ↾ {𝐵}):{𝐵}⟶𝐶 ↔ (𝐹‘𝐵) ∈ 𝐶)) |