Step | Hyp | Ref
| Expression |
1 | | funfn 4931 |
. 2
⊢ (Fun
𝐹 ↔ 𝐹 Fn dom 𝐹) |
2 | | elpreima 5286 |
. . . 4
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)))) |
3 | | elun 3084 |
. . . . . 6
⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵))) |
4 | | elpreima 5286 |
. . . . . . 7
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴))) |
5 | | elpreima 5286 |
. . . . . . 7
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐵) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) |
6 | 4, 5 | orbi12d 707 |
. . . . . 6
⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (◡𝐹 “ 𝐴) ∨ 𝑥 ∈ (◡𝐹 “ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))) |
7 | 3, 6 | syl5bb 181 |
. . . . 5
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵)))) |
8 | | elun 3084 |
. . . . . . 7
⊢ ((𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵) ↔ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)) |
9 | 8 | anbi2i 430 |
. . . . . 6
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ (𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵))) |
10 | | andi 731 |
. . . . . 6
⊢ ((𝑥 ∈ dom 𝐹 ∧ ((𝐹‘𝑥) ∈ 𝐴 ∨ (𝐹‘𝑥) ∈ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) |
11 | 9, 10 | bitri 173 |
. . . . 5
⊢ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∨ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐵))) |
12 | 7, 11 | syl6rbbr 188 |
. . . 4
⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))) |
13 | 2, 12 | bitrd 177 |
. . 3
⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ (𝐴 ∪ 𝐵)) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))) |
14 | 13 | eqrdv 2038 |
. 2
⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))) |
15 | 1, 14 | sylbi 114 |
1
⊢ (Fun
𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))) |