Step | Hyp | Ref
| Expression |
1 | | elfvdm 6130 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met) |
2 | | fmval 21557 |
. . . 4
⊢ ((𝑋 ∈ dom ∞Met ∧
𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))) |
3 | 1, 2 | syl3an1 1351 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))) |
4 | 3 | eleq1d 2672 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷))) |
5 | | simp1 1054 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
6 | | simp2 1055 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐵 ∈ (fBas‘𝑌)) |
7 | | simp3 1056 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋) |
8 | 1 | 3ad2ant1 1075 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ dom ∞Met) |
9 | | eqid 2610 |
. . . . 5
⊢ ran
(𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) |
10 | 9 | fbasrn 21498 |
. . . 4
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋 ∧ 𝑋 ∈ dom ∞Met) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋)) |
11 | 6, 7, 8, 10 | syl3anc 1318 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋)) |
12 | | fgcfil 22877 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) ∈ (fBas‘𝑋)) → ((𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥)) |
13 | 5, 11, 12 | syl2anc 691 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥)) |
14 | | imassrn 5396 |
. . . . . . . 8
⊢ (𝐹 “ 𝑦) ⊆ ran 𝐹 |
15 | | frn 5966 |
. . . . . . . . 9
⊢ (𝐹:𝑌⟶𝑋 → ran 𝐹 ⊆ 𝑋) |
16 | 15 | 3ad2ant3 1077 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ran 𝐹 ⊆ 𝑋) |
17 | 14, 16 | syl5ss 3579 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑦) ⊆ 𝑋) |
18 | 8, 17 | ssexd 4733 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐹 “ 𝑦) ∈ V) |
19 | 18 | ralrimivw 2950 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∀𝑦 ∈ 𝐵 (𝐹 “ 𝑦) ∈ V) |
20 | | eqid 2610 |
. . . . . 6
⊢ (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)) |
21 | | raleq 3115 |
. . . . . . 7
⊢ (𝑠 = (𝐹 “ 𝑦) → (∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥)) |
22 | 21 | raleqbi1dv 3123 |
. . . . . 6
⊢ (𝑠 = (𝐹 “ 𝑦) → (∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥)) |
23 | 20, 22 | rexrnmpt 6277 |
. . . . 5
⊢
(∀𝑦 ∈
𝐵 (𝐹 “ 𝑦) ∈ V → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥)) |
24 | 19, 23 | syl 17 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥)) |
25 | | simpl3 1059 |
. . . . . . . 8
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝐹:𝑌⟶𝑋) |
26 | | ffn 5958 |
. . . . . . . 8
⊢ (𝐹:𝑌⟶𝑋 → 𝐹 Fn 𝑌) |
27 | 25, 26 | syl 17 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝐹 Fn 𝑌) |
28 | | fbelss 21447 |
. . . . . . . 8
⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝑌) |
29 | 6, 28 | sylan 487 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝑌) |
30 | | oveq1 6556 |
. . . . . . . . . 10
⊢ (𝑢 = (𝐹‘𝑧) → (𝑢𝐷𝑣) = ((𝐹‘𝑧)𝐷𝑣)) |
31 | 30 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑢 = (𝐹‘𝑧) → ((𝑢𝐷𝑣) < 𝑥 ↔ ((𝐹‘𝑧)𝐷𝑣) < 𝑥)) |
32 | 31 | ralbidv 2969 |
. . . . . . . 8
⊢ (𝑢 = (𝐹‘𝑧) → (∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥)) |
33 | 32 | ralima 6402 |
. . . . . . 7
⊢ ((𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥)) |
34 | 27, 29, 33 | syl2anc 691 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥)) |
35 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑣 = (𝐹‘𝑤) → ((𝐹‘𝑧)𝐷𝑣) = ((𝐹‘𝑧)𝐷(𝐹‘𝑤))) |
36 | 35 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑣 = (𝐹‘𝑤) → (((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
37 | 36 | ralima 6402 |
. . . . . . . 8
⊢ ((𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌) → (∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
38 | 27, 29, 37 | syl2anc 691 |
. . . . . . 7
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
39 | 38 | ralbidv 2969 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑧 ∈ 𝑦 ∀𝑣 ∈ (𝐹 “ 𝑦)((𝐹‘𝑧)𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
40 | 34, 39 | bitrd 267 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑦 ∈ 𝐵) → (∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
41 | 40 | rexbidva 3031 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑦 ∈ 𝐵 ∀𝑢 ∈ (𝐹 “ 𝑦)∀𝑣 ∈ (𝐹 “ 𝑦)(𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
42 | 24, 41 | bitrd 267 |
. . 3
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
43 | 42 | ralbidv 2969 |
. 2
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑥 ∈ ℝ+ ∃𝑠 ∈ ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))∀𝑢 ∈ 𝑠 ∀𝑣 ∈ 𝑠 (𝑢𝐷𝑣) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |
44 | 4, 13, 43 | 3bitrd 293 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) |