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Mirrors > Home > MPE Home > Th. List > cmphmph | Structured version Visualization version GIF version |
Description: Compactness is a topological property-that is, for any two homeomorphic topologies, either both are compact or neither is. (Contributed by Jeff Hankins, 30-Jun-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
cmphmph | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmph 21949 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4159 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | eqid 2824 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2824 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 21937 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾) |
6 | f1ofo 6384 | . . . . . 6 ⊢ (𝑓:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝑓:∪ 𝐽–onto→∪ 𝐾) | |
7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–onto→∪ 𝐾) |
8 | hmeocn 21933 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
9 | 4 | cncmp 21565 | . . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp) |
10 | 9 | 3expb 1155 | . . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ (𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾))) → 𝐾 ∈ Comp) |
11 | 10 | expcom 404 | . . . . 5 ⊢ ((𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
12 | 7, 8, 11 | syl2anc 581 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
13 | 12 | exlimiv 2031 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
14 | 2, 13 | sylbi 209 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
15 | 1, 14 | sylbi 209 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∃wex 1880 ∈ wcel 2166 ≠ wne 2998 ∅c0 4143 ∪ cuni 4657 class class class wbr 4872 –onto→wfo 6120 –1-1-onto→wf1o 6121 (class class class)co 6904 Cn ccn 21398 Compccmp 21559 Homeochmeo 21926 ≃ chmph 21927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-reu 3123 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-fin 8225 df-top 21068 df-topon 21085 df-cn 21401 df-cmp 21560 df-hmeo 21928 df-hmph 21929 |
This theorem is referenced by: ptcmpfi 21986 xrcmp 23116 reheibor 34179 |
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