Theorem iincld 22765

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 2730	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 22755	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
3		dfss4 4259	. . . . . . 7 ⊢ (𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
4	2, 3	sylib 217	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
5	4	ralimi 3081	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
6		iineq2 5018	. . . . 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 480	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
9		iindif2 5081	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
10	9	adantr 479	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
11	8, 10	eqtr3d 2772	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
12		r19.2z 4495	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
13		cldrcl 22752	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
14	13	rexlimivw 3149	. . . 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 22757	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
17	16	ralimi 3081	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
18	17	adantl 480	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
19		iunopn 22622	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
20	15, 18, 19	syl2anc 582	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
21	1	opncld 22759	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
22	15, 20, 21	syl2anc 582	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
23	11, 22	eqeltrd 2831	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  ∪ cuni 4909  ∪ ciun 4998  ∩ ciin 4999  ‘cfv 6544  Topctop 22617  Clsdccld 22742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-top 22618  df-cld 22745

This theorem is referenced by:  intcld  22766  riincld  22770  hauscmplem  23132  ubthlem1  30388