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Mirrors > Home > MPE Home > Th. List > haushmph | Structured version Visualization version GIF version |
Description: Hausdorff-ness is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
haushmph | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | haustop 23322 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
2 | cnhaus 23345 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Haus) | |
3 | 1, 2 | haushmphlem 23778 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∪ cuni 4907 class class class wbr 5145 Hauscha 23299 ≃ chmph 23745 |
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 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 |
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 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-ov 7418 df-oprab 7419 df-mpo 7420 df-1st 7994 df-2nd 7995 df-1o 8487 df-map 8848 df-top 22883 df-topon 22900 df-cn 23218 df-haus 23306 df-hmeo 23746 df-hmph 23747 |
This theorem is referenced by: ishaus3 23814 |
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