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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 23700 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
2 | n0 4350 | . . 3 ⊢ ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾)) | |
3 | eqid 2728 | . . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
4 | eqid 2728 | . . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
5 | 3, 4 | hmeof1o 23688 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓:∪ 𝐽–1-1-onto→∪ 𝐾) |
6 | f1ofo 6851 | . . . . . 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 23684 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾)) | |
9 | 4 | cncmp 23316 | . . . . . . 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 1925 | . . 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 1773 ∈ wcel 2098 ≠ wne 2937 ∅c0 4326 ∪ cuni 4912 class class class wbr 5152 –onto→wfo 6551 –1-1-onto→wf1o 6552 (class class class)co 7426 Cn ccn 23148 Compccmp 23310 Homeochmeo 23677 ≃ chmph 23678 |
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 2699 ax-sep 5303 ax-nul 5310 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 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 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 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-fin 8974 df-top 22816 df-topon 22833 df-cn 23151 df-cmp 23311 df-hmeo 23679 df-hmph 23680 |
This theorem is referenced by: ptcmpfi 23737 xrcmp 24893 reheibor 37345 |
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