Theorem iincld 21641

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 2821	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 21631	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
3		dfss4 4234	. . . . . . 7 ⊢ (𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
4	2, 3	sylib 220	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
5	4	ralimi 3160	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
6		iineq2 4931	. . . . 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 484	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
9		iindif2 4991	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
10	9	adantr 483	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
11	8, 10	eqtr3d 2858	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
12		r19.2z 4439	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
13		cldrcl 21628	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
14	13	rexlimivw 3282	. . . 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 21633	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
17	16	ralimi 3160	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
18	17	adantl 484	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
19		iunopn 21500	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
20	15, 18, 19	syl2anc 586	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
21	1	opncld 21635	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
22	15, 20, 21	syl2anc 586	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
23	11, 22	eqeltrd 2913	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  ∪ cuni 4831  ∪ ciun 4911  ∩ ciin 4912  ‘cfv 6349  Topctop 21495  Clsdccld 21618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357  df-top 21496  df-cld 21621

This theorem is referenced by:  intcld  21642  riincld  21646  hauscmplem  22008  ubthlem1  28641