Theorem clsval2 22994

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 3400	. . . . . 6 ⊢ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)}
2		clscld.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
3	2	cldopn 22975	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑧) ∈ 𝐽)
4	3	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝐽)
5		sscon 4095	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
6	5	ad2antll 729	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
7	2	topopn 22850	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8		difexg 5274	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑧) ∈ V)
9		elpwg 4557	. . . . . . . . . . . . . 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 726	. . . . . . . . . . . 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 4152	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
14	2	cldss 22973	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (Clsd‘𝐽) → 𝑧 ⊆ 𝑋)
15	14	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 ⊆ 𝑋)
16		dfss4 4221	. . . . . . . . . . . 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 2742	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧)))
19		difeq2 4072	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ 𝑧) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑧)))
20	19	rspceeqv 3599	. . . . . . . . . 10 ⊢ (((𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ∧ 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧))) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥))
21	13, 18, 20	syl2anc 584	. . . . . . . . 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 4153	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝐽)
25	2	opncld 22977	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
26	23, 24, 25	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽))
27		elinel2 4154	. . . . . . . . . . . . . 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 4563	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ (𝑋 ∖ 𝑆))
30	29	difss2d 4091	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ 𝑋)
31		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ 𝑋)
32		ssconb 4094	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑆 ⊆ 𝑋) → (𝑥 ⊆ (𝑋 ∖ 𝑆) ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
33	30, 31, 32	syl2anc 584	. . . . . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽)))
37		sseq2 3960	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	36, 37	anbi12d 632	. . . . . . . . . 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 3137	. . . . . . . 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 2802	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
43	1, 42	eqtrid 2783	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
44	43	inteqd 4907	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
45		difexg 5274	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑥) ∈ V)
46	45	ralrimivw 3132	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V)
47		dfiin2g 4986	. . . . . 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 2774	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
51	2	clsval 22981	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧})
52		uniiun 5014	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) = ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥
53	52	difeq2i 4075	. . . . 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 22848	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
56	55	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝐽)
57		0elpw 5301	. . . . . . 7 ⊢ ∅ ∈ 𝒫 (𝑋 ∖ 𝑆)
58	57	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝒫 (𝑋 ∖ 𝑆))
59	56, 58	elind 4152	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
60		ne0i 4293	. . . . 5 ⊢ (∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅)
61		iindif2 5032	. . . . 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 2774	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
64	50, 51, 63	3eqtr4d 2781	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
65		difssd 4089	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ 𝑆) ⊆ 𝑋)
66	2	ntrval 22980	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ 𝑆) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
67	65, 66	sylan2 593	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ 𝑆)) = ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
68	67	difeq2d 4078	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
69	64, 68	eqtr4d 2774	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3399  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  ∪ cuni 4863  ∩ cint 4902  ∪ ciun 4946  ∩ ciin 4947  ‘cfv 6492  Topctop 22837  Clsdccld 22960  intcnt 22961  clsccl 22962

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-cld 22963  df-ntr 22964  df-cls 22965

This theorem is referenced by:  ntrval2  22995  clsdif  22997  cmclsopn  23006  bcth3  25287