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Mirrors > Home > MPE Home > Th. List > hmph | Structured version Visualization version GIF version |
Description: Express the predicate 𝐽 is homeomorphic to 𝐾. (Contributed by FL, 14-Feb-2007.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmph | ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hmph 22888 | . 2 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
2 | hmeofn 22889 | . 2 ⊢ Homeo Fn (Top × Top) | |
3 | 1, 2 | brwitnlem 8313 | 1 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ≠ wne 2944 ∅c0 4261 class class class wbr 5078 × cxp 5586 (class class class)co 7268 Topctop 22023 Homeochmeo 22885 ≃ chmph 22886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-fv 6438 df-ov 7271 df-oprab 7272 df-mpo 7273 df-1st 7817 df-2nd 7818 df-1o 8281 df-hmeo 22887 df-hmph 22888 |
This theorem is referenced by: hmphi 22909 hmphsym 22914 hmphtr 22915 hmphen 22917 haushmphlem 22919 cmphmph 22920 connhmph 22921 reghmph 22925 nrmhmph 22926 hmphdis 22928 hmphen2 22931 |
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