| Step | Hyp | Ref
| Expression |
| 1 | | reseq1 5991 |
. . 3
⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
| 2 | | reseq1 5991 |
. . 3
⊢ (𝐹 = 𝐺 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
| 3 | 1, 2 | jca 511 |
. 2
⊢ (𝐹 = 𝐺 → ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))) |
| 4 | | elun 4153 |
. . . . 5
⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) |
| 5 | | fveq1 6905 |
. . . . . . . . 9
⊢ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑥) = ((𝐺 ↾ 𝐴)‘𝑥)) |
| 6 | | fvres 6925 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
| 7 | 5, 6 | sylan9req 2798 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → ((𝐺 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥)) |
| 8 | | fvres 6925 |
. . . . . . . . 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 2779 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 11 | 10 | adantlr 715 |
. . . . . 6
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 12 | | fveq1 6905 |
. . . . . . . . 9
⊢ ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) → ((𝐹 ↾ 𝐵)‘𝑥) = ((𝐺 ↾ 𝐵)‘𝑥)) |
| 13 | | fvres 6925 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
| 14 | 12, 13 | sylan9req 2798 |
. . . . . . . 8
⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝐺 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥)) |
| 15 | | fvres 6925 |
. . . . . . . . 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 2779 |
. . . . . . 7
⊢ (((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 18 | 17 | adantll 714 |
. . . . . 6
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 19 | 11, 18 | jaodan 960 |
. . . . 5
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 20 | 4, 19 | sylan2b 594 |
. . . 4
⊢ ((((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) ∧ 𝑥 ∈ (𝐴 ∪ 𝐵)) → (𝐹‘𝑥) = (𝐺‘𝑥)) |
| 21 | 20 | ralrimiva 3146 |
. . 3
⊢ (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥)) |
| 22 | | eqfnfv 7051 |
. . 3
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ (𝐴 ∪ 𝐵)(𝐹‘𝑥) = (𝐺‘𝑥))) |
| 23 | 21, 22 | imbitrrid 246 |
. 2
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) → 𝐹 = 𝐺)) |
| 24 | 3, 23 | impbid2 226 |
1
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))) |