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Theorem hmeocld 23791
Description: Homeomorphisms preserve closedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1 𝑋 = 𝐽
Assertion
Ref Expression
hmeocld ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))

Proof of Theorem hmeocld
StepHypRef Expression
1 hmeocnvcn 23785 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
21adantr 480 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
3 imacnvcnv 6228 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
4 cnclima 23292 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
53, 4eqeltrrid 2844 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
65ex 412 . . 3 (𝐹 ∈ (𝐾 Cn 𝐽) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹𝐴) ∈ (Clsd‘𝐾)))
72, 6syl 17 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹𝐴) ∈ (Clsd‘𝐾)))
8 hmeocn 23784 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
98adantr 480 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10 cnclima 23292 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ∈ (Clsd‘𝐾)) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽))
1110ex 412 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽)))
129, 11syl 17 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽)))
13 hmeoopn.1 . . . . . . 7 𝑋 = 𝐽
14 eqid 2735 . . . . . . 7 𝐾 = 𝐾
1513, 14hmeof1o 23788 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
16 f1of1 6848 . . . . . 6 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
1715, 16syl 17 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1 𝐾)
18 f1imacnv 6865 . . . . 5 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1917, 18sylan 580 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
2019eleq1d 2824 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
2112, 20sylibd 239 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
227, 21impbid 212 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wss 3963   cuni 4912  ccnv 5688  cima 5692  1-1wf1 6560  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Clsdccld 23040   Cn ccn 23248  Homeochmeo 23777
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-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-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  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-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-map 8867  df-top 22916  df-topon 22933  df-cld 23043  df-cn 23251  df-hmeo 23779
This theorem is referenced by:  cldsubg  24135  reheibor  37826
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