Theorem difopn 21645

Description: The difference of a closed set with an open set is open. (Contributed by Mario Carneiro, 6-Jan-2014.)

Hypothesis

Ref	Expression
iscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
difopn	⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)

Proof of Theorem difopn

Step	Hyp	Ref	Expression
1		elssuni 4871	. . . . . 6 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽)
2		iscld.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
3	1, 2	sseqtrrdi 4021	. . . . 5 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋)
4	3	adantr 483	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ⊆ 𝑋)
5		df-ss 3955	. . . 4 ⊢ (𝐴 ⊆ 𝑋 ↔ (𝐴 ∩ 𝑋) = 𝐴)
6	4, 5	sylib 220	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ 𝑋) = 𝐴)
7	6	difeq1d 4101	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) = (𝐴 ∖ 𝐵))
8		indif2 4250	. . 3 ⊢ (𝐴 ∩ (𝑋 ∖ 𝐵)) = ((𝐴 ∩ 𝑋) ∖ 𝐵)
9		cldrcl 21637	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
10	9	adantl 484	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐽 ∈ Top)
11		simpl 485	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → 𝐴 ∈ 𝐽)
12	2	cldopn 21642	. . . . 5 ⊢ (𝐵 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝐵) ∈ 𝐽)
13	12	adantl 484	. . . 4 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝐵) ∈ 𝐽)
14		inopn 21510	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽 ∧ (𝑋 ∖ 𝐵) ∈ 𝐽) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
15	10, 11, 13, 14	syl3anc 1367	. . 3 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∩ (𝑋 ∖ 𝐵)) ∈ 𝐽)
16	8, 15	eqeltrrid 2921	. 2 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → ((𝐴 ∩ 𝑋) ∖ 𝐵) ∈ 𝐽)
17	7, 16	eqeltrrd 2917	1 ⊢ ((𝐴 ∈ 𝐽 ∧ 𝐵 ∈ (Clsd‘𝐽)) → (𝐴 ∖ 𝐵) ∈ 𝐽)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ∖ cdif 3936   ∩ cin 3938   ⊆ wss 3939  ∪ cuni 4841  ‘cfv 6358  Topctop 21504  Clsdccld 21627

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 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366  df-top 21505  df-cld 21630

This theorem is referenced by:  bcthlem5  23934  cldssbrsiga  31450  pibt2  34702  poimirlem30  34926  dirkercncflem2  42396  fourierdlem62  42460