Theorem iincld 23068

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 2740	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	cldss 23058	. . . . . . 7 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐵 ⊆ ∪ 𝐽)
3		dfss4 4288	. . . . . . 7 ⊢ (𝐵 ⊆ ∪ 𝐽 ↔ (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
4	2, 3	sylib 218	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
5	4	ralimi 3089	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ (∪ 𝐽 ∖ 𝐵)) = 𝐵)
6		iineq2 5035	. . . . 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 5100	. . . 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 2782	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 = (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)))
12		r19.2z 4518	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∃𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))
13		cldrcl 23055	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
14	13	rexlimivw 3157	. . . 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 23060	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
17	16	ralimi 3089	. . . . 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 22925	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
20	15, 18, 19	syl2anc 583	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽)
21	1	opncld 23062	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵) ∈ 𝐽) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
22	15, 20, 21	syl2anc 583	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → (∪ 𝐽 ∖ ∪ 𝑥 ∈ 𝐴 (∪ 𝐽 ∖ 𝐵)) ∈ (Clsd‘𝐽))
23	11, 22	eqeltrd 2844	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽)) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067  ∃wrex 3076   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  ∪ cuni 4931  ∪ ciun 5015  ∩ ciin 5016  ‘cfv 6573  Topctop 22920  Clsdccld 23045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-top 22921  df-cld 23048

This theorem is referenced by:  intcld  23069  riincld  23073  hauscmplem  23435  ubthlem1  30902