Theorem hmeocld 21570

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) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))

Proof of Theorem hmeocld

Step	Hyp	Ref	Expression
1		hmeocnvcn 21564	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
2	1	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
3		imacnvcnv 5599	. . . . 5 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
4		cnclima 21072	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡◡𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
5	3, 4	syl5eqelr 2706	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
6	5	ex 450	. . 3 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
7	2, 6	syl 17	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
8		hmeocn 21563	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
9	8	adantr 481	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		cnclima 21072	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽))
11	10	ex 450	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽)))
12	9, 11	syl 17	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽)))
13		hmeoopn.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
14		eqid 2622	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
15	13, 14	hmeof1o 21567	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
16		f1of1 6136	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
17	15, 16	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1→∪ 𝐾)
18		f1imacnv 6153	. . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
19	17, 18	sylan 488	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
20	19	eleq1d 2686	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
21	12, 20	sylibd 229	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
22	7, 21	impbid 202	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∪ cuni 4436  ^◡ccnv 5113   “ cima 5117  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Clsdccld 20820   Cn ccn 21028  Homeochmeo 21556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-cld 20823  df-cn 21031  df-hmeo 21558

This theorem is referenced by:  cldsubg  21914  reheibor  33638