Step | Hyp | Ref
| Expression |
1 | | reseq1 5311 |
. . 3
⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
2 | | reseq1 5311 |
. . 3
⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
3 | 1, 2 | jca 553 |
. 2
⊢ (𝐹 = 𝐺 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))) |
4 | | elun 3715 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) |
5 | | fveq1 6102 |
. . . . . . . . 9
⊢ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐺 ↾ 𝐴)‘𝑥)) |
6 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
7 | 5, 6 | sylan9req 2665 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
8 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥)) |
9 | 8 | adantl 481 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐺‘𝑥)) |
10 | 7, 9 | eqtr3d 2646 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
11 | 10 | adantlr 747 |
. . . . . 6
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
12 | | fveq1 6102 |
. . . . . . . . 9
⊢ ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥)) |
13 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
14 | 12, 13 | sylan9req 2665 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
15 | | fvres 6117 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥)) |
16 | 15 | adantl 481 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐺‘𝑥)) |
17 | 14, 16 | eqtr3d 2646 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
18 | 17 | adantll 746 |
. . . . . 6
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
19 | 11, 18 | jaodan 822 |
. . . . 5
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
20 | 4, 19 | sylan2b 491 |
. . . 4
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
21 | 20 | ralrimiva 2949 |
. . 3
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥)) |
22 | | eqfnfv 6219 |
. . 3
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥))) |
23 | 21, 22 | syl5ibr 235 |
. 2
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → 𝐹 = 𝐺)) |
24 | 3, 23 | impbid2 215 |
1
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))) |