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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 23771 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4349 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | eqid 2726 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2726 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 23759 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾) |
6 | f1ofo 6850 | . . . . . 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 23755 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
9 | 4 | cncmp 23387 | . . . . . . 7 ⊢ ((𝐽 ∈ Comp ∧ 𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp) |
10 | 9 | 3expb 1117 | . . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ (𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾))) → 𝐾 ∈ Comp) |
11 | 10 | expcom 412 | . . . . 5 ⊢ ((𝑓:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
12 | 7, 8, 11 | syl2anc 582 | . . . 4 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
13 | 12 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
14 | 2, 13 | sylbi 216 | . 2 ⊢ ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
15 | 1, 14 | sylbi 216 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ∅c0 4325 ∪ cuni 4913 class class class wbr 5153 –onto→wfo 6552 –1-1-onto→wf1o 6553 (class class class)co 7424 Cn ccn 23219 Compccmp 23381 Homeochmeo 23748 ≃ chmph 23749 |
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 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-1o 8496 df-map 8857 df-en 8975 df-dom 8976 df-fin 8978 df-top 22887 df-topon 22904 df-cn 23222 df-cmp 23382 df-hmeo 23750 df-hmph 23751 |
This theorem is referenced by: ptcmpfi 23808 xrcmp 24964 reheibor 37540 |
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