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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 23704 | . 2 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
2 | hmeofn 23705 | . 2 ⊢ Homeo Fn (Top × Top) | |
3 | 1, 2 | brwitnlem 8528 | 1 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ≠ wne 2929 ∅c0 4322 class class class wbr 5149 × cxp 5676 (class class class)co 7419 Topctop 22839 Homeochmeo 23701 ≃ chmph 23702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-1o 8487 df-hmeo 23703 df-hmph 23704 |
This theorem is referenced by: hmphi 23725 hmphsym 23730 hmphtr 23731 hmphen 23733 haushmphlem 23735 cmphmph 23736 connhmph 23737 reghmph 23741 nrmhmph 23742 hmphdis 23744 hmphen2 23747 |
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