| Step | Hyp | Ref
| Expression |
| 1 | | ulmrel 26421 |
. . . 4
⊢ Rel
(⇝𝑢‘𝑆) |
| 2 | 1 | brrelex12i 5740 |
. . 3
⊢ (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V)) |
| 3 | 2 | a1i 11 |
. 2
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
| 4 | | 3simpa 1149 |
. . . 4
⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ)) |
| 5 | | fvex 6919 |
. . . . . . 7
⊢
(ℤ≥‘𝑛) ∈ V |
| 6 | | fex 7246 |
. . . . . . 7
⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
(ℤ≥‘𝑛) ∈ V) → 𝐹 ∈ V) |
| 7 | 5, 6 | mpan2 691 |
. . . . . 6
⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆)
→ 𝐹 ∈
V) |
| 8 | 7 | a1i 11 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆)
→ 𝐹 ∈
V)) |
| 9 | | fex 7246 |
. . . . . 6
⊢ ((𝐺:𝑆⟶ℂ ∧ 𝑆 ∈ 𝑉) → 𝐺 ∈ V) |
| 10 | 9 | expcom 413 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 → (𝐺:𝑆⟶ℂ → 𝐺 ∈ V)) |
| 11 | 8, 10 | anim12d 609 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
| 12 | 4, 11 | syl5 34 |
. . 3
⊢ (𝑆 ∈ 𝑉 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
| 13 | 12 | rexlimdvw 3160 |
. 2
⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) → (𝐹 ∈ V ∧ 𝐺 ∈ V))) |
| 14 | | df-ulm 26420 |
. . . . . 6
⊢
⇝𝑢 = (𝑠 ∈ V ↦ {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑠) ∧
𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
| 15 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (ℂ ↑m 𝑠) = (ℂ ↑m
𝑆)) |
| 16 | 15 | feq3d 6723 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑠)
↔ 𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆))) |
| 17 | | feq2 6717 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (𝑦:𝑠⟶ℂ ↔ 𝑦:𝑆⟶ℂ)) |
| 18 | | raleq 3323 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑆 → (∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
| 19 | 18 | rexralbidv 3223 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑆 → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
| 20 | 19 | ralbidv 3178 |
. . . . . . . . 9
⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) |
| 21 | 16, 17, 20 | 3anbi123d 1438 |
. . . . . . . 8
⊢ (𝑠 = 𝑆 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑠) ∧
𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))) |
| 22 | 21 | rexbidv 3179 |
. . . . . . 7
⊢ (𝑠 = 𝑆 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑠) ∧
𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥))) |
| 23 | 22 | opabbidv 5209 |
. . . . . 6
⊢ (𝑠 = 𝑆 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑠) ∧
𝑦:𝑠⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑠 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
| 24 | | elex 3501 |
. . . . . 6
⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) |
| 25 | | simpr1 1195 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆)) |
| 26 | | uzssz 12899 |
. . . . . . . . . . . . 13
⊢
(ℤ≥‘𝑛) ⊆ ℤ |
| 27 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ (ℂ
↑m 𝑆)
∈ V |
| 28 | | zex 12622 |
. . . . . . . . . . . . . 14
⊢ ℤ
∈ V |
| 29 | | elpm2r 8885 |
. . . . . . . . . . . . . 14
⊢
((((ℂ ↑m 𝑆) ∈ V ∧ ℤ ∈ V) ∧
(𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
(ℤ≥‘𝑛) ⊆ ℤ)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm
ℤ)) |
| 30 | 27, 28, 29 | mpanl12 702 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
(ℤ≥‘𝑛) ⊆ ℤ) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm
ℤ)) |
| 31 | 25, 26, 30 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm
ℤ)) |
| 32 | | simpr2 1196 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦:𝑆⟶ℂ) |
| 33 | | cnex 11236 |
. . . . . . . . . . . . . 14
⊢ ℂ
∈ V |
| 34 | | simpl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑆 ∈ 𝑉) |
| 35 | | elmapg 8879 |
. . . . . . . . . . . . . 14
⊢ ((ℂ
∈ V ∧ 𝑆 ∈
𝑉) → (𝑦 ∈ (ℂ
↑m 𝑆)
↔ 𝑦:𝑆⟶ℂ)) |
| 36 | 33, 34, 35 | sylancr 587 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑦 ∈ (ℂ ↑m 𝑆) ↔ 𝑦:𝑆⟶ℂ)) |
| 37 | 32, 36 | mpbird 257 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → 𝑦 ∈ (ℂ ↑m 𝑆)) |
| 38 | 31, 37 | jca 511 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ 𝑉 ∧ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)
∧ 𝑦 ∈ (ℂ
↑m 𝑆))) |
| 39 | 38 | ex 412 |
. . . . . . . . . 10
⊢ (𝑆 ∈ 𝑉 → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)
∧ 𝑦 ∈ (ℂ
↑m 𝑆)))) |
| 40 | 39 | rexlimdvw 3160 |
. . . . . . . . 9
⊢ (𝑆 ∈ 𝑉 → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) → (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)
∧ 𝑦 ∈ (ℂ
↑m 𝑆)))) |
| 41 | 40 | ssopab2dv 5556 |
. . . . . . . 8
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ {〈𝑓, 𝑦〉 ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)
∧ 𝑦 ∈ (ℂ
↑m 𝑆))}) |
| 42 | | df-xp 5691 |
. . . . . . . 8
⊢
(((ℂ ↑m 𝑆) ↑pm ℤ) ×
(ℂ ↑m 𝑆)) = {〈𝑓, 𝑦〉 ∣ (𝑓 ∈ ((ℂ ↑m 𝑆) ↑pm ℤ)
∧ 𝑦 ∈ (ℂ
↑m 𝑆))} |
| 43 | 41, 42 | sseqtrrdi 4025 |
. . . . . . 7
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ ↑m
𝑆) ↑pm
ℤ) × (ℂ ↑m 𝑆))) |
| 44 | | ovex 7464 |
. . . . . . . . 9
⊢ ((ℂ
↑m 𝑆)
↑pm ℤ) ∈ V |
| 45 | 44, 27 | xpex 7773 |
. . . . . . . 8
⊢
(((ℂ ↑m 𝑆) ↑pm ℤ) ×
(ℂ ↑m 𝑆)) ∈ V |
| 46 | 45 | ssex 5321 |
. . . . . . 7
⊢
({〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ⊆ (((ℂ ↑m
𝑆) ↑pm
ℤ) × (ℂ ↑m 𝑆)) → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V) |
| 47 | 43, 46 | syl 17 |
. . . . . 6
⊢ (𝑆 ∈ 𝑉 → {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} ∈ V) |
| 48 | 14, 23, 24, 47 | fvmptd3 7039 |
. . . . 5
⊢ (𝑆 ∈ 𝑉 →
(⇝𝑢‘𝑆) = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}) |
| 49 | 48 | breqd 5154 |
. . . 4
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ 𝐹{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺)) |
| 50 | | simpl 482 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑓 = 𝐹) |
| 51 | 50 | feq1d 6720 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆)
↔ 𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆))) |
| 52 | | simpr 484 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → 𝑦 = 𝐺) |
| 53 | 52 | feq1d 6720 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦:𝑆⟶ℂ ↔ 𝐺:𝑆⟶ℂ)) |
| 54 | 50 | fveq1d 6908 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑓‘𝑘) = (𝐹‘𝑘)) |
| 55 | 54 | fveq1d 6908 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓‘𝑘)‘𝑧) = ((𝐹‘𝑘)‘𝑧)) |
| 56 | 52 | fveq1d 6908 |
. . . . . . . . . . . . 13
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (𝑦‘𝑧) = (𝐺‘𝑧)) |
| 57 | 55, 56 | oveq12d 7449 |
. . . . . . . . . . . 12
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧)) = (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) |
| 58 | 57 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) = (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)))) |
| 59 | 58 | breq1d 5153 |
. . . . . . . . . 10
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
| 60 | 59 | ralbidv 3178 |
. . . . . . . . 9
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
| 61 | 60 | rexralbidv 3223 |
. . . . . . . 8
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
| 62 | 61 | ralbidv 3178 |
. . . . . . 7
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) |
| 63 | 51, 53, 62 | 3anbi123d 1438 |
. . . . . 6
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
| 64 | 63 | rexbidv 3179 |
. . . . 5
⊢ ((𝑓 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
| 65 | | eqid 2737 |
. . . . 5
⊢
{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} = {〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)} |
| 66 | 64, 65 | brabga 5539 |
. . . 4
⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹{〈𝑓, 𝑦〉 ∣ ∃𝑛 ∈ ℤ (𝑓:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝑦:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝑓‘𝑘)‘𝑧) − (𝑦‘𝑧))) < 𝑥)}𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
| 67 | 49, 66 | sylan9bb 509 |
. . 3
⊢ ((𝑆 ∈ 𝑉 ∧ (𝐹 ∈ V ∧ 𝐺 ∈ V)) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |
| 68 | 67 | ex 412 |
. 2
⊢ (𝑆 ∈ 𝑉 → ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))) |
| 69 | 3, 13, 68 | pm5.21ndd 379 |
1
⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ
↑m 𝑆) ∧
𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑗 ∈
(ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))) |