Theorem iincld 21936

Description: The indexed intersection of a collection 𝐵(𝑥) of closed sets is closed. Theorem 6.1(2) of [Munkres] p. 93. (Contributed by NM, 5-Oct-2006.) (Revised by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
iincld	⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iincld

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 21926	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
3		dfss4 4173	. . . . . . 7 ⊢ (𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
4	2, 3	sylib 221	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
5	4	ralimi 3083	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
6		iineq2 4924	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵 → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
7	5, 6	syl 17	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
8	7	adantl 485	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
9		iindif2 4985	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
10	9	adantr 484	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
11	8, 10	eqtr3d 2779	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
12		r19.2z 4406	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
13		cldrcl 21923	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
14	13	rexlimivw 3201	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
15	12, 14	syl 17	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
16	1	cldopn 21928	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
17	16	ralimi 3083	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
18	17	adantl 485	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
19		iunopn 21795	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
20	15, 18, 19	syl2anc 587	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
21	1	opncld 21930	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
22	15, 20, 21	syl2anc 587	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
23	11, 22	eqeltrd 2838	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062   ∖ cdif 3863   ⊆ wss 3866  ∅c0 4237  ∪ cuni 4819  ∪ ciun 4904  ∩ ciin 4905  ‘cfv 6380  Topctop 21790  Clsdccld 21913

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-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-iun 4906  df-iin 4907  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-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-top 21791  df-cld 21916

This theorem is referenced by:  intcld  21937  riincld  21941  hauscmplem  22303  ubthlem1  28951