| Step | Hyp | Ref
| Expression |
| 1 | | eluni 4910 |
. 2
⊢ (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)) |
| 2 | | funfn 6596 |
. . . . . . . 8
⊢ (Fun
𝐹 ↔ 𝐹 Fn dom 𝐹) |
| 3 | | fvelrnb 6969 |
. . . . . . . 8
⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)) |
| 4 | 2, 3 | sylbi 217 |
. . . . . . 7
⊢ (Fun
𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)) |
| 5 | 4 | anbi2d 630 |
. . . . . 6
⊢ (Fun
𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))) |
| 6 | | r19.42v 3191 |
. . . . . 6
⊢
(∃𝑥 ∈ dom
𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)) |
| 7 | 5, 6 | bitr4di 289 |
. . . . 5
⊢ (Fun
𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦))) |
| 8 | | eleq2 2830 |
. . . . . . 7
⊢ ((𝐹‘𝑥) = 𝑦 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ 𝑦)) |
| 9 | 8 | biimparc 479 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → 𝐴 ∈ (𝐹‘𝑥)) |
| 10 | 9 | reximi 3084 |
. . . . 5
⊢
(∃𝑥 ∈ dom
𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)) |
| 11 | 7, 10 | biimtrdi 253 |
. . . 4
⊢ (Fun
𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
| 12 | 11 | exlimdv 1933 |
. . 3
⊢ (Fun
𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
| 13 | | fvelrn 7096 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
| 14 | 13 | a1d 25 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ ran 𝐹)) |
| 15 | 14 | ancld 550 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹))) |
| 16 | | fvex 6919 |
. . . . . 6
⊢ (𝐹‘𝑥) ∈ V |
| 17 | | eleq2 2830 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑥) → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ (𝐹‘𝑥))) |
| 18 | | eleq1 2829 |
. . . . . . 7
⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹)) |
| 19 | 17, 18 | anbi12d 632 |
. . . . . 6
⊢ (𝑦 = (𝐹‘𝑥) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹))) |
| 20 | 16, 19 | spcev 3606 |
. . . . 5
⊢ ((𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)) |
| 21 | 15, 20 | syl6 35 |
. . . 4
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))) |
| 22 | 21 | rexlimdva 3155 |
. . 3
⊢ (Fun
𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))) |
| 23 | 12, 22 | impbid 212 |
. 2
⊢ (Fun
𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |
| 24 | 1, 23 | bitrid 283 |
1
⊢ (Fun
𝐹 → (𝐴 ∈ ∪ ran
𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))) |