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Mirrors > Home > MPE Home > Th. List > hmeocnvb | 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, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmeocnvb | ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmeocnv 23041 | . . 3 ⊢ (◡𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐾Homeo𝐽)) | |
2 | dfrel2 6138 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | eleq1 2826 | . . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) | |
4 | 2, 3 | sylbi 216 | . . 3 ⊢ (Rel 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
5 | 1, 4 | imbitrid 243 | . 2 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))) |
6 | hmeocnv 23041 | . 2 ⊢ (𝐹 ∈ (𝐾Homeo𝐽) → ◡𝐹 ∈ (𝐽Homeo𝐾)) | |
7 | 5, 6 | impbid1 224 | 1 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ◡ccnv 5630 Rel wrel 5636 (class class class)co 7350 Homeochmeo 23032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7353 df-oprab 7354 df-mpo 7355 df-map 8701 df-top 22171 df-topon 22188 df-cn 22506 df-hmeo 23034 |
This theorem is referenced by: (None) |
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