Step | Hyp | Ref
| Expression |
1 | | ismtyhmeolem.5 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ (𝑀 Ismty 𝑁)) |
2 | | ismtyhmeolem.3 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) |
3 | | ismtyhmeolem.4 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) |
4 | | isismty 35955 |
. . . . . 6
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))) |
5 | 2, 3, 4 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))) |
6 | 1, 5 | mpbid 231 |
. . . 4
⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) |
7 | 6 | simpld 495 |
. . 3
⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) |
8 | | f1of 6714 |
. . 3
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌) |
9 | 7, 8 | syl 17 |
. 2
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) |
10 | 3 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌)) |
11 | 2 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋)) |
12 | | ismtycnv 35956 |
. . . . . . . . . 10
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀))) |
13 | 2, 3, 12 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹 ∈ (𝑀 Ismty 𝑁) → ◡𝐹 ∈ (𝑁 Ismty 𝑀))) |
14 | 1, 13 | mpd 15 |
. . . . . . . 8
⊢ (𝜑 → ◡𝐹 ∈ (𝑁 Ismty 𝑀)) |
15 | 14 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)) |
16 | | simprl 768 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑤 ∈ 𝑌) |
17 | | simprr 770 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈
ℝ*) |
18 | | ismtyima 35957 |
. . . . . . 7
⊢ (((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑀 ∈ (∞Met‘𝑋) ∧ ◡𝐹 ∈ (𝑁 Ismty 𝑀)) ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) = ((◡𝐹‘𝑤)(ball‘𝑀)𝑟)) |
19 | 10, 11, 15, 16, 17, 18 | syl32anc 1377 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) = ((◡𝐹‘𝑤)(ball‘𝑀)𝑟)) |
20 | | f1ocnv 6726 |
. . . . . . . . 9
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) |
21 | | f1of 6714 |
. . . . . . . . 9
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋) |
22 | 7, 20, 21 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → ◡𝐹:𝑌⟶𝑋) |
23 | | simpl 483 |
. . . . . . . 8
⊢ ((𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*) → 𝑤 ∈ 𝑌) |
24 | | ffvelrn 6956 |
. . . . . . . 8
⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑤 ∈ 𝑌) → (◡𝐹‘𝑤) ∈ 𝑋) |
25 | 22, 23, 24 | syl2an 596 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹‘𝑤) ∈ 𝑋) |
26 | | ismtyhmeo.1 |
. . . . . . . 8
⊢ 𝐽 = (MetOpen‘𝑀) |
27 | 26 | blopn 23654 |
. . . . . . 7
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (◡𝐹‘𝑤) ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → ((◡𝐹‘𝑤)(ball‘𝑀)𝑟) ∈ 𝐽) |
28 | 11, 25, 17, 27 | syl3anc 1370 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → ((◡𝐹‘𝑤)(ball‘𝑀)𝑟) ∈ 𝐽) |
29 | 19, 28 | eqeltrd 2841 |
. . . . 5
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽) |
30 | 29 | ralrimivva 3117 |
. . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝑌 ∀𝑟 ∈ ℝ* (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽) |
31 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((ball‘𝑁)‘𝑧) = ((ball‘𝑁)‘〈𝑤, 𝑟〉)) |
32 | | df-ov 7274 |
. . . . . . . 8
⊢ (𝑤(ball‘𝑁)𝑟) = ((ball‘𝑁)‘〈𝑤, 𝑟〉) |
33 | 31, 32 | eqtr4di 2798 |
. . . . . . 7
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((ball‘𝑁)‘𝑧) = (𝑤(ball‘𝑁)𝑟)) |
34 | 33 | imaeq2d 5968 |
. . . . . 6
⊢ (𝑧 = 〈𝑤, 𝑟〉 → (◡𝐹 “ ((ball‘𝑁)‘𝑧)) = (◡𝐹 “ (𝑤(ball‘𝑁)𝑟))) |
35 | 34 | eleq1d 2825 |
. . . . 5
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽 ↔ (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽)) |
36 | 35 | ralxp 5749 |
. . . 4
⊢
(∀𝑧 ∈
(𝑌 ×
ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽 ↔ ∀𝑤 ∈ 𝑌 ∀𝑟 ∈ ℝ* (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽) |
37 | 30, 36 | sylibr 233 |
. . 3
⊢ (𝜑 → ∀𝑧 ∈ (𝑌 × ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽) |
38 | | blf 23558 |
. . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → (ball‘𝑁):(𝑌 ×
ℝ*)⟶𝒫 𝑌) |
39 | | ffn 6598 |
. . . 4
⊢
((ball‘𝑁):(𝑌 ×
ℝ*)⟶𝒫 𝑌 → (ball‘𝑁) Fn (𝑌 ×
ℝ*)) |
40 | | imaeq2 5964 |
. . . . . 6
⊢ (𝑢 = ((ball‘𝑁)‘𝑧) → (◡𝐹 “ 𝑢) = (◡𝐹 “ ((ball‘𝑁)‘𝑧))) |
41 | 40 | eleq1d 2825 |
. . . . 5
⊢ (𝑢 = ((ball‘𝑁)‘𝑧) → ((◡𝐹 “ 𝑢) ∈ 𝐽 ↔ (◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽)) |
42 | 41 | ralrn 6961 |
. . . 4
⊢
((ball‘𝑁) Fn
(𝑌 ×
ℝ*) → (∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽 ↔ ∀𝑧 ∈ (𝑌 × ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽)) |
43 | 3, 38, 39, 42 | 4syl 19 |
. . 3
⊢ (𝜑 → (∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽 ↔ ∀𝑧 ∈ (𝑌 × ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽)) |
44 | 37, 43 | mpbird 256 |
. 2
⊢ (𝜑 → ∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽) |
45 | 26 | mopntopon 23590 |
. . . 4
⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
46 | 2, 45 | syl 17 |
. . 3
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
47 | | ismtyhmeo.2 |
. . . . 5
⊢ 𝐾 = (MetOpen‘𝑁) |
48 | 47 | mopnval 23589 |
. . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝑁))) |
49 | 3, 48 | syl 17 |
. . 3
⊢ (𝜑 → 𝐾 = (topGen‘ran (ball‘𝑁))) |
50 | 47 | mopntopon 23590 |
. . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌)) |
51 | 3, 50 | syl 17 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
52 | 46, 49, 51 | tgcn 22401 |
. 2
⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽))) |
53 | 9, 44, 52 | mpbir2and 710 |
1
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |