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Theorem cmphmph 23812
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.)
Assertion
Ref Expression
cmphmph (𝐽𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))

Proof of Theorem cmphmph
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 hmph 23800 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4359 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 eqid 2735 . . . . . . 7 𝐽 = 𝐽
4 eqid 2735 . . . . . . 7 𝐾 = 𝐾
53, 4hmeof1o 23788 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
6 f1ofo 6856 . . . . . 6 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐽onto 𝐾)
75, 6syl 17 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽onto 𝐾)
8 hmeocn 23784 . . . . 5 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
94cncmp 23416 . . . . . . 7 ((𝐽 ∈ Comp ∧ 𝑓: 𝐽onto 𝐾𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
1093expb 1119 . . . . . 6 ((𝐽 ∈ Comp ∧ (𝑓: 𝐽onto 𝐾𝑓 ∈ (𝐽 Cn 𝐾))) → 𝐾 ∈ Comp)
1110expcom 413 . . . . 5 ((𝑓: 𝐽onto 𝐾𝑓 ∈ (𝐽 Cn 𝐾)) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
127, 8, 11syl2anc 584 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
1312exlimiv 1928 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
142, 13sylbi 217 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
151, 14sylbi 217 1 (𝐽𝐾 → (𝐽 ∈ Comp → 𝐾 ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1776  wcel 2106  wne 2938  c0 4339   cuni 4912   class class class wbr 5148  ontowfo 6561  1-1-ontowf1o 6562  (class class class)co 7431   Cn ccn 23248  Compccmp 23410  Homeochmeo 23777  chmph 23778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-en 8985  df-dom 8986  df-fin 8988  df-top 22916  df-topon 22933  df-cn 23251  df-cmp 23411  df-hmeo 23779  df-hmph 23780
This theorem is referenced by:  ptcmpfi  23837  xrcmp  24993  reheibor  37826
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