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Theorem hmeocld 23116
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 23110 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
21adantr 481 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
3 imacnvcnv 6158 . . . . 5 (𝐹𝐴) = (𝐹𝐴)
4 cnclima 22617 . . . . 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 413 . . 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 23109 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
98adantr 481 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10 cnclima 22617 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ∈ (Clsd‘𝐾)) → (𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽))
1110ex 413 . . . 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 2736 . . . . . . 7 𝐾 = 𝐾
1513, 14hmeof1o 23113 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
16 f1of1 6783 . . . . . 6 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
1715, 16syl 17 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1 𝐾)
18 f1imacnv 6800 . . . . 5 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1917, 18sylan 580 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
2019eleq1d 2822 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹 “ (𝐹𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
2112, 20sylibd 238 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
227, 21impbid 211 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹𝐴) ∈ (Clsd‘𝐾)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wss 3910   cuni 4865  ccnv 5632  cima 5636  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7356  Clsdccld 22365   Cn ccn 22573  Homeochmeo 23102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8766  df-top 22241  df-topon 22258  df-cld 22368  df-cn 22576  df-hmeo 23104
This theorem is referenced by:  cldsubg  23460  reheibor  36289
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