Step | Hyp | Ref
| Expression |
1 | | eufnfv.1 |
. . . . 5
⊢ 𝐴 ∈ V |
2 | 1 | mptex 6486 |
. . . 4
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V |
3 | | eqeq2 2633 |
. . . . . 6
⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 = 𝑧 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))) |
4 | 3 | bibi2d 332 |
. . . . 5
⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))) |
5 | 4 | albidv 1849 |
. . . 4
⊢ (𝑧 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧) ↔ ∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)))) |
6 | 2, 5 | spcev 3300 |
. . 3
⊢
(∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) → ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧)) |
7 | | eufnfv.2 |
. . . . . . 7
⊢ 𝐵 ∈ V |
8 | | eqid 2622 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) |
9 | 7, 8 | fnmpti 6022 |
. . . . . 6
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 |
10 | | fneq1 5979 |
. . . . . 6
⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → (𝑓 Fn 𝐴 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)) |
11 | 9, 10 | mpbiri 248 |
. . . . 5
⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) → 𝑓 Fn 𝐴) |
12 | 11 | pm4.71ri 665 |
. . . 4
⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵))) |
13 | | dffn5 6241 |
. . . . . . 7
⊢ (𝑓 Fn 𝐴 ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥))) |
14 | | eqeq1 2626 |
. . . . . . 7
⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵))) |
15 | 13, 14 | sylbi 207 |
. . . . . 6
⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵))) |
16 | | fvex 6201 |
. . . . . . . 8
⊢ (𝑓‘𝑥) ∈ V |
17 | 16 | rgenw 2924 |
. . . . . . 7
⊢
∀𝑥 ∈
𝐴 (𝑓‘𝑥) ∈ V |
18 | | mpteqb 6299 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐴 (𝑓‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)) |
19 | 17, 18 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) |
20 | 15, 19 | syl6bb 276 |
. . . . 5
⊢ (𝑓 Fn 𝐴 → (𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)) |
21 | 20 | pm5.32i 669 |
. . . 4
⊢ ((𝑓 Fn 𝐴 ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)) |
22 | 12, 21 | bitr2i 265 |
. . 3
⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ 𝐵)) |
23 | 6, 22 | mpg 1724 |
. 2
⊢
∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧) |
24 | | df-eu 2474 |
. 2
⊢
(∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ ∃𝑧∀𝑓((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) ↔ 𝑓 = 𝑧)) |
25 | 23, 24 | mpbir 221 |
1
⊢
∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵) |