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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 22363 | . 2 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
2 | hmeofn 22364 | . 2 ⊢ Homeo Fn (Top × Top) | |
3 | 1, 2 | brwitnlem 8131 | 1 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ≠ wne 3016 ∅c0 4290 class class class wbr 5065 × cxp 5552 (class class class)co 7155 Topctop 21500 Homeochmeo 22360 ≃ chmph 22361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-1o 8101 df-hmeo 22362 df-hmph 22363 |
This theorem is referenced by: hmphi 22384 hmphsym 22389 hmphtr 22390 hmphen 22392 haushmphlem 22394 cmphmph 22395 connhmph 22396 reghmph 22400 nrmhmph 22401 hmphdis 22403 hmphen2 22406 |
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