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Theorem hmeocld 22664
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 22658 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
21adantr 484 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
3 imacnvcnv 6069 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
4 cnclima 22165 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
53, 4eqeltrrid 2843 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
65ex 416 . . 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 22657 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
98adantr 484 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10 cnclima 22165 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ∈ (Clsd‘𝐾)) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽))
1110ex 416 . . . 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 22661 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
16 f1of1 6660 . . . . . 6 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
1715, 16syl 17 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1 𝐾)
18 f1imacnv 6677 . . . . 5 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1917, 18sylan 583 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
2019eleq1d 2822 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
2112, 20sylibd 242 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
227, 21impbid 215 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wss 3866   cuni 4819  ccnv 5550  cima 5554  1-1wf1 6377  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  Clsdccld 21913   Cn ccn 22121  Homeochmeo 22650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-map 8510  df-top 21791  df-topon 21808  df-cld 21916  df-cn 22124  df-hmeo 22652
This theorem is referenced by:  cldsubg  23008  reheibor  35734
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