Proof of Theorem cldregopn
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | opnregcld.1 | . . . . 5
⊢ 𝑋 = ∪
𝐽 | 
| 2 | 1 | clscld 23055 | . . . 4
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽)) | 
| 3 |  | eqcom 2744 | . . . . 5
⊢
(((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) | 
| 4 | 3 | biimpi 216 | . . . 4
⊢
(((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) | 
| 5 |  | fveq2 6906 | . . . . 5
⊢ (𝑐 = ((cls‘𝐽)‘𝐴) → ((int‘𝐽)‘𝑐) = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) | 
| 6 | 5 | rspceeqv 3645 | . . . 4
⊢
((((cls‘𝐽)‘𝐴) ∈ (Clsd‘𝐽) ∧ 𝐴 = ((int‘𝐽)‘((cls‘𝐽)‘𝐴))) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)) | 
| 7 | 2, 4, 6 | syl2an 596 | . . 3
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴) → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐)) | 
| 8 | 7 | ex 412 | . 2
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 → ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))) | 
| 9 |  | cldrcl 23034 | . . . . . . 7
⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top) | 
| 10 | 1 | cldss 23037 | . . . . . . 7
⊢ (𝑐 ∈ (Clsd‘𝐽) → 𝑐 ⊆ 𝑋) | 
| 11 | 1 | ntrss2 23065 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑐) | 
| 12 | 9, 10, 11 | syl2anc 584 | . . . . . . . 8
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑐) | 
| 13 | 1 | clsss2 23080 | . . . . . . . 8
⊢ ((𝑐 ∈ (Clsd‘𝐽) ∧ ((int‘𝐽)‘𝑐) ⊆ 𝑐) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) | 
| 14 | 12, 13 | mpdan 687 | . . . . . . 7
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) | 
| 15 | 1 | ntrss 23063 | . . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋 ∧ ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐)) | 
| 16 | 9, 10, 14, 15 | syl3anc 1373 | . . . . . 6
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) ⊆ ((int‘𝐽)‘𝑐)) | 
| 17 | 1 | ntridm 23076 | . . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐)) | 
| 18 | 9, 10, 17 | syl2anc 584 | . . . . . . 7
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) = ((int‘𝐽)‘𝑐)) | 
| 19 | 1 | ntrss3 23068 | . . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑐 ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ 𝑋) | 
| 20 | 9, 10, 19 | syl2anc 584 | . . . . . . . . 9
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ 𝑋) | 
| 21 | 1 | clsss3 23067 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧
((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋) | 
| 22 | 9, 20, 21 | syl2anc 584 | . . . . . . . 8
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋) | 
| 23 | 1 | sscls 23064 | . . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧
((int‘𝐽)‘𝑐) ⊆ 𝑋) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) | 
| 24 | 9, 20, 23 | syl2anc 584 | . . . . . . . 8
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) | 
| 25 | 1 | ntrss 23063 | . . . . . . . 8
⊢ ((𝐽 ∈ Top ∧
((cls‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ 𝑋 ∧ ((int‘𝐽)‘𝑐) ⊆ ((cls‘𝐽)‘((int‘𝐽)‘𝑐))) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐)))) | 
| 26 | 9, 22, 24, 25 | syl3anc 1373 | . . . . . . 7
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑐)) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐)))) | 
| 27 | 18, 26 | eqsstrrd 4019 | . . . . . 6
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘𝑐) ⊆ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐)))) | 
| 28 | 16, 27 | eqssd 4001 | . . . . 5
⊢ (𝑐 ∈ (Clsd‘𝐽) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)) | 
| 29 | 28 | adantl 481 | . . . 4
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐)) | 
| 30 |  | 2fveq3 6911 | . . . . 5
⊢ (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐)))) | 
| 31 |  | id 22 | . . . . 5
⊢ (𝐴 = ((int‘𝐽)‘𝑐) → 𝐴 = ((int‘𝐽)‘𝑐)) | 
| 32 | 30, 31 | eqeq12d 2753 | . . . 4
⊢ (𝐴 = ((int‘𝐽)‘𝑐) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ((int‘𝐽)‘((cls‘𝐽)‘((int‘𝐽)‘𝑐))) = ((int‘𝐽)‘𝑐))) | 
| 33 | 29, 32 | syl5ibrcom 247 | . . 3
⊢ (((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) ∧ 𝑐 ∈ (Clsd‘𝐽)) → (𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴)) | 
| 34 | 33 | rexlimdva 3155 | . 2
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐) → ((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴)) | 
| 35 | 8, 34 | impbid 212 | 1
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (((int‘𝐽)‘((cls‘𝐽)‘𝐴)) = 𝐴 ↔ ∃𝑐 ∈ (Clsd‘𝐽)𝐴 = ((int‘𝐽)‘𝑐))) |