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Theorem hmeocld 23775
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 23769 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
21adantr 480 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
3 imacnvcnv 6226 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
4 cnclima 23276 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
53, 4eqeltrrid 2846 . . . 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 23768 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
98adantr 480 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10 cnclima 23276 . . . . 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 2737 . . . . . . 7 𝐾 = 𝐾
1513, 14hmeof1o 23772 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
16 f1of1 6847 . . . . . 6 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
1715, 16syl 17 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1 𝐾)
18 f1imacnv 6864 . . . . 5 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1917, 18sylan 580 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
2019eleq1d 2826 . . 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 1540  wcel 2108  wss 3951   cuni 4907  ccnv 5684  cima 5688  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Clsdccld 23024   Cn ccn 23232  Homeochmeo 23761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cld 23027  df-cn 23235  df-hmeo 23763
This theorem is referenced by:  cldsubg  24119  reheibor  37846
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