Theorem iincld 23004

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 2736	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 22994	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
3		dfss4 4209	. . . . . . 7 ⊢ (𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
4	2, 3	sylib 218	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
5	4	ralimi 3074	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
6		iineq2 4954	. . . . 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 481	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = ∩ 𝑥 ∈ 𝐴 𝐵)
9		iindif2 5019	. . . 4 ⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
10	9	adantr 480	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
11	8, 10	eqtr3d 2773	. 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 22991	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
14	13	rexlimivw 3134	. . . 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 22996	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
17	16	ralimi 3074	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
18	17	adantl 481	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
19		iunopn 22863	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
20	15, 18, 19	syl2anc 585	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
21	1	opncld 22998	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
22	15, 20, 21	syl2anc 585	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
23	11, 22	eqeltrd 2836	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850  ∪ ciun 4933  ∩ ciin 4934  ‘cfv 6498  Topctop 22858  Clsdccld 22981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-top 22859  df-cld 22984

This theorem is referenced by:  intcld  23005  riincld  23009  hauscmplem  23371  ubthlem1  30941