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Mirrors > Home > MPE Home > Th. List > hmeocnv | Structured version Visualization version GIF version |
Description: The converse of a homeomorphism is a homeomorphism. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmeocnv | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnvcn 22368 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
2 | hmeocn 22367 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
3 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2821 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 21853 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
6 | frel 6518 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹) | |
7 | 2, 5, 6 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹) |
8 | dfrel2 6045 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
9 | 7, 8 | sylib 220 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 = 𝐹) |
10 | 9, 2 | eqeltrd 2913 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐽 Cn 𝐾)) |
11 | ishmeo 22366 | . 2 ⊢ (◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ◡◡𝐹 ∈ (𝐽 Cn 𝐾))) | |
12 | 1, 10, 11 | sylanbrc 585 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cuni 4837 ◡ccnv 5553 Rel wrel 5559 ⟶wf 6350 (class class class)co 7155 Cn ccn 21831 Homeochmeo 22360 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-top 21501 df-topon 21518 df-cn 21834 df-hmeo 22362 |
This theorem is referenced by: hmeocnvb 22381 hmphsym 22389 xpstopnlem2 22418 |
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