Theorem hmeocld 21793

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 21787	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
2	1	adantr 472	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
3		imacnvcnv 5758	. . . . 5 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
4		cnclima 21295	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡◡𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
5	3, 4	syl5eqelr 2845	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
6	5	ex 449	. . 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 21786	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
9	8	adantr 472	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		cnclima 21295	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽))
11	10	ex 449	. . . 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 2761	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
15	13, 14	hmeof1o 21790	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
16		f1of1 6299	. . . . . 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 6316	. . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
19	17, 18	sylan 489	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
20	19	eleq1d 2825	. . 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 383   = wceq 1632   ∈ wcel 2140   ⊆ wss 3716  ∪ cuni 4589  ^◡ccnv 5266   “ cima 5270  –1-1→wf1 6047  –1-1-onto→wf1o 6049  ‘cfv 6050  (class class class)co 6815  Clsdccld 21043   Cn ccn 21251  Homeochmeo 21779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-top 20922  df-topon 20939  df-cld 21046  df-cn 21254  df-hmeo 21781

This theorem is referenced by:  cldsubg  22136  reheibor  33970