| 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 37808 | . . . . . 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 232 | . . . 4
⊢ (𝜑 → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))) | 
| 7 | 6 | simpld 494 | . . 3
⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌) | 
| 8 |  | f1of 6848 | . . 3
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌) | 
| 9 | 7, 8 | syl 17 | . 2
⊢ (𝜑 → 𝐹:𝑋⟶𝑌) | 
| 10 | 3 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌)) | 
| 11 | 2 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋)) | 
| 12 |  | ismtycnv 37809 | . . . . . . . . . 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 480 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → ◡𝐹 ∈ (𝑁 Ismty 𝑀)) | 
| 16 |  | simprl 771 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑤 ∈ 𝑌) | 
| 17 |  | simprr 773 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈
ℝ*) | 
| 18 |  | ismtyima 37810 | . . . . . . 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 1380 | . . . . . 6
⊢ ((𝜑 ∧ (𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*)) → (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) = ((◡𝐹‘𝑤)(ball‘𝑀)𝑟)) | 
| 20 |  | f1ocnv 6860 | . . . . . . . . 9
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) | 
| 21 |  | f1of 6848 | . . . . . . . . 9
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋) | 
| 22 | 7, 20, 21 | 3syl 18 | . . . . . . . 8
⊢ (𝜑 → ◡𝐹:𝑌⟶𝑋) | 
| 23 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑤 ∈ 𝑌 ∧ 𝑟 ∈ ℝ*) → 𝑤 ∈ 𝑌) | 
| 24 |  | ffvelcdm 7101 | . . . . . . . 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 24513 | . . . . . . 7
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (◡𝐹‘𝑤) ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → ((◡𝐹‘𝑤)(ball‘𝑀)𝑟) ∈ 𝐽) | 
| 28 | 11, 25, 17, 27 | syl3anc 1373 | . . . . . 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 3202 | . . . 4
⊢ (𝜑 → ∀𝑤 ∈ 𝑌 ∀𝑟 ∈ ℝ* (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽) | 
| 31 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((ball‘𝑁)‘𝑧) = ((ball‘𝑁)‘〈𝑤, 𝑟〉)) | 
| 32 |  | df-ov 7434 | . . . . . . . 8
⊢ (𝑤(ball‘𝑁)𝑟) = ((ball‘𝑁)‘〈𝑤, 𝑟〉) | 
| 33 | 31, 32 | eqtr4di 2795 | . . . . . . 7
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((ball‘𝑁)‘𝑧) = (𝑤(ball‘𝑁)𝑟)) | 
| 34 | 33 | imaeq2d 6078 | . . . . . 6
⊢ (𝑧 = 〈𝑤, 𝑟〉 → (◡𝐹 “ ((ball‘𝑁)‘𝑧)) = (◡𝐹 “ (𝑤(ball‘𝑁)𝑟))) | 
| 35 | 34 | eleq1d 2826 | . . . . 5
⊢ (𝑧 = 〈𝑤, 𝑟〉 → ((◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽 ↔ (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽)) | 
| 36 | 35 | ralxp 5852 | . . . 4
⊢
(∀𝑧 ∈
(𝑌 ×
ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽 ↔ ∀𝑤 ∈ 𝑌 ∀𝑟 ∈ ℝ* (◡𝐹 “ (𝑤(ball‘𝑁)𝑟)) ∈ 𝐽) | 
| 37 | 30, 36 | sylibr 234 | . . 3
⊢ (𝜑 → ∀𝑧 ∈ (𝑌 × ℝ*)(◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽) | 
| 38 |  | blf 24417 | . . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → (ball‘𝑁):(𝑌 ×
ℝ*)⟶𝒫 𝑌) | 
| 39 |  | ffn 6736 | . . . 4
⊢
((ball‘𝑁):(𝑌 ×
ℝ*)⟶𝒫 𝑌 → (ball‘𝑁) Fn (𝑌 ×
ℝ*)) | 
| 40 |  | imaeq2 6074 | . . . . . 6
⊢ (𝑢 = ((ball‘𝑁)‘𝑧) → (◡𝐹 “ 𝑢) = (◡𝐹 “ ((ball‘𝑁)‘𝑧))) | 
| 41 | 40 | eleq1d 2826 | . . . . 5
⊢ (𝑢 = ((ball‘𝑁)‘𝑧) → ((◡𝐹 “ 𝑢) ∈ 𝐽 ↔ (◡𝐹 “ ((ball‘𝑁)‘𝑧)) ∈ 𝐽)) | 
| 42 | 41 | ralrn 7108 | . . . 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 257 | . 2
⊢ (𝜑 → ∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽) | 
| 45 | 26 | mopntopon 24449 | . . . 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 24448 | . . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 = (topGen‘ran (ball‘𝑁))) | 
| 49 | 3, 48 | syl 17 | . . 3
⊢ (𝜑 → 𝐾 = (topGen‘ran (ball‘𝑁))) | 
| 50 | 47 | mopntopon 24449 | . . . 4
⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌)) | 
| 51 | 3, 50 | syl 17 | . . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) | 
| 52 | 46, 49, 51 | tgcn 23260 | . 2
⊢ (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑢 ∈ ran (ball‘𝑁)(◡𝐹 “ 𝑢) ∈ 𝐽))) | 
| 53 | 9, 44, 52 | mpbir2and 713 | 1
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |