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Mirrors > Home > MPE Home > Th. List > hmeocld | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoopn.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hmeocld | ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnvcn 23615 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) |
3 | imacnvcnv 6198 | . . . . 5 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴) | |
4 | cnclima 23122 | . . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡◡𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) | |
5 | 3, 4 | eqeltrrid 2832 | . . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) |
6 | 5 | ex 412 | . . 3 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) |
7 | 2, 6 | syl 17 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) |
8 | hmeocn 23614 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾)) |
10 | cnclima 23122 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽)) | |
11 | 10 | ex 412 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽))) |
12 | 9, 11 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽))) |
13 | hmeoopn.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
14 | eqid 2726 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
15 | 13, 14 | hmeof1o 23618 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾) |
16 | f1of1 6825 | . . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾) | |
17 | 15, 16 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1→∪ 𝐾) |
18 | f1imacnv 6842 | . . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴) | |
19 | 17, 18 | sylan 579 | . . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴) |
20 | 19 | eleq1d 2812 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽))) |
21 | 12, 20 | sylibd 238 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽))) |
22 | 7, 21 | impbid 211 | 1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3943 ∪ cuni 4902 ◡ccnv 5668 “ cima 5672 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 (class class class)co 7404 Clsdccld 22870 Cn ccn 23078 Homeochmeo 23607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-top 22746 df-topon 22763 df-cld 22873 df-cn 23081 df-hmeo 23609 |
This theorem is referenced by: cldsubg 23965 reheibor 37219 |
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