![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23785 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
2 | hmeocn 23784 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
3 | eqid 2735 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2735 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 23270 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
6 | frel 6742 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹) | |
7 | 2, 5, 6 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹) |
8 | dfrel2 6211 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
9 | 7, 8 | sylib 218 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 = 𝐹) |
10 | 9, 2 | eqeltrd 2839 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐽 Cn 𝐾)) |
11 | ishmeo 23783 | . 2 ⊢ (◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ◡◡𝐹 ∈ (𝐽 Cn 𝐾))) | |
12 | 1, 10, 11 | sylanbrc 583 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 ◡ccnv 5688 Rel wrel 5694 ⟶wf 6559 (class class class)co 7431 Cn ccn 23248 Homeochmeo 23777 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8867 df-top 22916 df-topon 22933 df-cn 23251 df-hmeo 23779 |
This theorem is referenced by: hmeocnvb 23798 hmphsym 23806 xpstopnlem2 23835 |
Copyright terms: Public domain | W3C validator |