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Mirrors > Home > ILE Home > Th. List > hmeocnvb | 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 12479 | . . 3 ⊢ (◡𝐹 ∈ (𝐽Homeo𝐾) → ◡◡𝐹 ∈ (𝐾Homeo𝐽)) | |
2 | dfrel2 4989 | . . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹) | |
3 | eleq1 2202 | . . . 4 ⊢ (◡◡𝐹 = 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) | |
4 | 2, 3 | sylbi 120 | . . 3 ⊢ (Rel 𝐹 → (◡◡𝐹 ∈ (𝐾Homeo𝐽) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
5 | 1, 4 | syl5ib 153 | . 2 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾Homeo𝐽))) |
6 | hmeocnv 12479 | . 2 ⊢ (𝐹 ∈ (𝐾Homeo𝐽) → ◡𝐹 ∈ (𝐽Homeo𝐾)) | |
7 | 5, 6 | impbid1 141 | 1 ⊢ (Rel 𝐹 → (◡𝐹 ∈ (𝐽Homeo𝐾) ↔ 𝐹 ∈ (𝐾Homeo𝐽))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 ◡ccnv 4538 Rel wrel 4544 (class class class)co 5774 Homeochmeo 12472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12168 df-topon 12181 df-cn 12360 df-hmeo 12473 |
This theorem is referenced by: (None) |
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