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Mirrors > Home > MPE Home > Th. List > hmphsym | Structured version Visualization version GIF version |
Description: "Is homeomorphic to" is symmetric. (Contributed by FL, 8-Mar-2007.) (Proof shortened by Mario Carneiro, 30-May-2014.) |
Ref | Expression |
---|---|
hmphsym | ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmph 23766 | . . 3 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4347 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | 1, 2 | bitri 274 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) |
4 | hmeocnv 23752 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → ◡𝑓 ∈ (𝐾Homeo𝐽)) | |
5 | hmphi 23767 | . . . 4 ⊢ (◡𝑓 ∈ (𝐾Homeo𝐽) → 𝐾 ≃ 𝐽) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ≃ 𝐽) |
7 | 6 | exlimiv 1926 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ≃ 𝐽) |
8 | 3, 7 | sylbi 216 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∅c0 4323 class class class wbr 5144 ◡ccnv 5672 (class class class)co 7414 Homeochmeo 23743 ≃ chmph 23744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-1o 8486 df-map 8847 df-top 22882 df-topon 22899 df-cn 23217 df-hmeo 23745 df-hmph 23746 |
This theorem is referenced by: hmpher 23774 hmphsymb 23776 haushmphlem 23777 t0kq 23808 kqhmph 23809 ist1-5lem 23810 reheibor 37551 |
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