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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 21543 | . 2 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) | |
2 | cnhaus 21566 | . 2 ⊢ ((𝐽 ∈ Haus ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Haus) | |
3 | 1, 2 | haushmphlem 21999 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Haus → 𝐾 ∈ Haus)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∪ cuni 4671 class class class wbr 4886 Hauscha 21520 ≃ chmph 21966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-1st 7445 df-2nd 7446 df-1o 7843 df-map 8142 df-top 21106 df-topon 21123 df-cn 21439 df-haus 21527 df-hmeo 21967 df-hmph 21968 |
This theorem is referenced by: ishaus3 22035 |
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