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Mirrors > Home > MPE Home > Th. List > hmeoopn | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
hmeoopn.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hmeoopn | ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeoima 22367 | . . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾) | |
2 | 1 | ex 415 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐴 ∈ 𝐽 → (𝐹 “ 𝐴) ∈ 𝐾)) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 → (𝐹 “ 𝐴) ∈ 𝐾)) |
4 | hmeocn 22362 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
5 | cnima 21867 | . . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ 𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽) | |
6 | 5 | ex 415 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽)) |
7 | 4, 6 | syl 17 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽)) |
8 | 7 | adantr 483 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽)) |
9 | hmeoopn.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
10 | eqid 2821 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
11 | 9, 10 | hmeof1o 22366 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾) |
12 | f1of1 6608 | . . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1→∪ 𝐾) |
14 | f1imacnv 6625 | . . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴) | |
15 | 13, 14 | sylan 582 | . . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴) |
16 | 15 | eleq1d 2897 | . . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽 ↔ 𝐴 ∈ 𝐽)) |
17 | 8, 16 | sylibd 241 | . 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ 𝐾 → 𝐴 ∈ 𝐽)) |
18 | 3, 17 | impbid 214 | 1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ∪ cuni 4831 ◡ccnv 5548 “ cima 5552 –1-1→wf1 6346 –1-1-onto→wf1o 6348 (class class class)co 7150 Cn ccn 21826 Homeochmeo 22355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-top 21496 df-topon 21513 df-cn 21829 df-hmeo 22357 |
This theorem is referenced by: hmphdis 22398 |
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