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Theorem hmeocld 21380
 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 21374 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
21adantr 480 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
3 imacnvcnv 5517 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
4 cnclima 20882 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
53, 4syl5eqelr 2693 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹𝐴) ∈ (Clsd‘𝐾))
65ex 449 . . 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 21373 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
98adantr 480 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10 cnclima 20882 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ∈ (Clsd‘𝐾)) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽))
1110ex 449 . . . 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 2610 . . . . . . 7 𝐾 = 𝐾
1513, 14hmeof1o 21377 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
16 f1of1 6049 . . . . . 6 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
1715, 16syl 17 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1 𝐾)
18 f1imacnv 6066 . . . . 5 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1917, 18sylan 487 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
2019eleq1d 2672 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
2112, 20sylibd 228 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
227, 21impbid 201 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037   “ cima 5041  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Clsdccld 20630   Cn ccn 20838  Homeochmeo 21366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-cld 20633  df-cn 20841  df-hmeo 21368 This theorem is referenced by:  cldsubg  21724  reheibor  32808
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