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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 21655 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽)) | |
2 | hmeocn 21654 | . . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
3 | eqid 2692 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2692 | . . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | cnf 21141 | . . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
6 | frel 6131 | . . . . 5 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → Rel 𝐹) | |
7 | 2, 5, 6 | 3syl 18 | . . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → Rel 𝐹) |
8 | dfrel2 5661 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
9 | 7, 8 | sylib 208 | . . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 = 𝐹) |
10 | 9, 2 | eqeltrd 2771 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐽 Cn 𝐾)) |
11 | ishmeo 21653 | . 2 ⊢ (◡𝐹 ∈ (𝐾Homeo𝐽) ↔ (◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ ◡◡𝐹 ∈ (𝐽 Cn 𝐾))) | |
12 | 1, 10, 11 | sylanbrc 701 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾Homeo𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1564 ∈ wcel 2071 ∪ cuni 4512 ◡ccnv 5185 Rel wrel 5191 ⟶wf 5965 (class class class)co 6733 Cn ccn 21119 Homeochmeo 21647 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1818 ax-5 1920 ax-6 1986 ax-7 2022 ax-8 2073 ax-9 2080 ax-10 2100 ax-11 2115 ax-12 2128 ax-13 2323 ax-ext 2672 ax-sep 4857 ax-nul 4865 ax-pow 4916 ax-pr 4979 ax-un 7034 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1567 df-ex 1786 df-nf 1791 df-sb 1979 df-eu 2543 df-mo 2544 df-clab 2679 df-cleq 2685 df-clel 2688 df-nfc 2823 df-ne 2865 df-ral 2987 df-rex 2988 df-rab 2991 df-v 3274 df-sbc 3510 df-dif 3651 df-un 3653 df-in 3655 df-ss 3662 df-nul 3992 df-if 4163 df-pw 4236 df-sn 4254 df-pr 4256 df-op 4260 df-uni 4513 df-br 4729 df-opab 4789 df-mpt 4806 df-id 5096 df-xp 5192 df-rel 5193 df-cnv 5194 df-co 5195 df-dm 5196 df-rn 5197 df-res 5198 df-ima 5199 df-iota 5932 df-fun 5971 df-fn 5972 df-f 5973 df-fv 5977 df-ov 6736 df-oprab 6737 df-mpt2 6738 df-map 7944 df-top 20790 df-topon 20807 df-cn 21122 df-hmeo 21649 |
This theorem is referenced by: hmeocnvb 21668 hmphsym 21676 xpstopnlem2 21705 |
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