Theorem clsval2 23006

Description: Express closure in terms of interior. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
clscld.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
clsval2	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Proof of Theorem clsval2

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3402	. . . . . 6 ⊢ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)}
2		clscld.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
3	2	cldopn 22987	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑧) ∈ 𝐽)
4	3	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝐽)
5		sscon 4097	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
6	5	ad2antll 730	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
7	2	topopn 22862	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8		difexg 5276	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑧) ∈ V)
9		elpwg 4559	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∖ 𝑧) ∈ V → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
10	7, 8, 9	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ Top → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
11	10	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
12	6, 11	mpbird 257	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆))
13	4, 12	elind 4154	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
14	2	cldss 22985	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (Clsd‘𝐽) → 𝑧 ⊆ 𝑋)
15	14	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 ⊆ 𝑋)
16		dfss4 4223	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
17	15, 16	sylib 218	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
18	17	eqcomd 2743	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧)))
19		difeq2 4074	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ 𝑧) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑧)))
20	19	rspceeqv 3601	. . . . . . . . . 10 ⊢ (((𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥))
21	13, 18, 20	syl2anc 585	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥))
22	21	ex 412	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
23		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
24		elinel1 4155	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝐽)
25	2	opncld 22989	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
26	23, 24, 25	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
27		elinel2 4156	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
28	27	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
29	28	elpwid 4565	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ (𝑋 ∖ 𝑆))
30	29	difss2d 4093	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ 𝑋)
31		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ 𝑋)
32		ssconb 4096	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑋) → (𝑥 ⊆ (𝑋 ∖ 𝑆) ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
33	30, 31, 32	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑥 ⊆ (𝑋 ∖ 𝑆) ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
34	29, 33	mpbid 232	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
35	26, 34	jca 511	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
36		eleq1 2825	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽)))
37		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	36, 37	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥))))
39	35, 38	syl5ibrcom 247	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
40	39	rexlimdva 3139	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
41	22, 40	impbid 212	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
42	41	abbidv 2803	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
43	1, 42	eqtrid 2784	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
44	43	inteqd 4909	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
45		difexg 5276	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑥) ∈ V)
46	45	ralrimivw 3134	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V)
47		dfiin2g 4988	. . . . . 6 ⊢ (∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
48	7, 46, 47	3syl 18	. . . . 5 ⊢ (𝐽 ∈ Top → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
49	48	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
50	44, 49	eqtr4d 2775	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
51	2	clsval 22993	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧})
52		uniiun 5016	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) = ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥
53	52	difeq2i 4077	. . . . 5 ⊢ (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥)
54	53	a1i 11	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
55		0opn 22860	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
56	55	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝐽)
57		0elpw 5303	. . . . . . 7 ⊢ ∅ ∈ 𝒫 (𝑋 ∖ 𝑆)
58	57	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝒫 (𝑋 ∖ 𝑆))
59	56, 58	elind 4154	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
60		ne0i 4295	. . . . 5 ⊢ (∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅)
61		iindif2 5034	. . . . 5 ⊢ ((𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅ → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
62	59, 60, 61	3syl 18	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) = (𝑋 ∖ ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥))
63	54, 62	eqtr4d 2775	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
64	50, 51, 63	3eqtr4d 2782	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
65		difssd 4091	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ 𝑆) ⊆ 𝑋)
66	2	ntrval 22992	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
67	65, 66	sylan2 594	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
68	67	difeq2d 4080	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
69	64, 68	eqtr4d 2775	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∖ cdif 3900   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865  ∩ cint 4904  ∪ ciun 4948  ∩ ciin 4949  ‘cfv 6500  Topctop 22849  Clsdccld 22972  intcnt 22973  clsccl 22974

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-top 22850  df-cld 22975  df-ntr 22976  df-cls 22977

This theorem is referenced by:  ntrval2  23007  clsdif  23009  cmclsopn  23018  bcth3  25299