Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22367 | . . 3 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4296 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | 1, 2 | bitri 277 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) |
4 | hmeocnv 22353 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → ◡𝑓 ∈ (𝐾Homeo𝐽)) | |
5 | hmphi 22368 | . . . 4 ⊢ (◡𝑓 ∈ (𝐾Homeo𝐽) → 𝐾 ≃ 𝐽) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ≃ 𝐽) |
7 | 6 | exlimiv 1931 | . 2 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → 𝐾 ≃ 𝐽) |
8 | 3, 7 | sylbi 219 | 1 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∅c0 4279 class class class wbr 5052 ◡ccnv 5540 (class class class)co 7142 Homeochmeo 22344 ≃ chmph 22345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-1o 8088 df-map 8394 df-top 21485 df-topon 21502 df-cn 21818 df-hmeo 22346 df-hmph 22347 |
This theorem is referenced by: hmpher 22375 hmphsymb 22377 haushmphlem 22378 t0kq 22409 kqhmph 22410 ist1-5lem 22411 reheibor 35149 |
Copyright terms: Public domain | W3C validator |