Theorem clsval2 22201

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 3073	. . . . . 6 ⊢ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)}
2		clscld.1	. . . . . . . . . . . . 13 ⊢ 𝑋 = ∪ 𝐽
3	2	cldopn 22182	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑧) ∈ 𝐽)
4	3	ad2antrl 725	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝐽)
5		sscon 4073	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ 𝑧 → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
6	5	ad2antll 726	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆))
7	2	topopn 22055	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8		difexg 5251	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑧) ∈ V)
9		elpwg 4536	. . . . . . . . . . . . . 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 723	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → ((𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆) ↔ (𝑋 ∖ 𝑧) ⊆ (𝑋 ∖ 𝑆)))
12	6, 11	mpbird 256	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ 𝒫 (𝑋 ∖ 𝑆))
13	4, 12	elind 4128	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ 𝑧) ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
14	2	cldss 22180	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (Clsd‘𝐽) → 𝑧 ⊆ 𝑋)
15	14	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 ⊆ 𝑋)
16		dfss4 4192	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
17	15, 16	sylib 217	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → (𝑋 ∖ (𝑋 ∖ 𝑧)) = 𝑧)
18	17	eqcomd 2744	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)) → 𝑧 = (𝑋 ∖ (𝑋 ∖ 𝑧)))
19		difeq2 4051	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑋 ∖ 𝑧) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑧)))
20	19	rspceeqv 3575	. . . . . . . . . 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 413	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) → ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
23		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝐽 ∈ Top)
24		elinel1 4129	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝐽)
25	2	opncld 22184	. . . . . . . . . . . 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 4130	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
28	27	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ∈ 𝒫 (𝑋 ∖ 𝑆))
29	28	elpwid 4544	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ (𝑋 ∖ 𝑆))
30	29	difss2d 4069	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑥 ⊆ 𝑋)
31		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ 𝑋)
32		ssconb 4072	. . . . . . . . . . . . 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 231	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → 𝑆 ⊆ (𝑋 ∖ 𝑥))
35	26, 34	jca 512	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
36		eleq1 2826	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ↔ (𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽)))
37		sseq2 3947	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → (𝑆 ⊆ 𝑧 ↔ 𝑆 ⊆ (𝑋 ∖ 𝑥)))
38	36, 37	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑧 = (𝑋 ∖ 𝑥) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ((𝑋 ∖ 𝑥) ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ (𝑋 ∖ 𝑥))))
39	35, 38	syl5ibrcom 246	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) → (𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
40	39	rexlimdva 3213	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥) → (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)))
41	22, 40	impbid 211	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧) ↔ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)))
42	41	abbidv 2807	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∣ (𝑧 ∈ (Clsd‘𝐽) ∧ 𝑆 ⊆ 𝑧)} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
43	1, 42	eqtrid 2790	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
44	43	inteqd 4884	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ {𝑧 ∣ ∃𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑧 = (𝑋 ∖ 𝑥)})
45		difexg 5251	. . . . . . 7 ⊢ (𝑋 ∈ 𝐽 → (𝑋 ∖ 𝑥) ∈ V)
46	45	ralrimivw 3104	. . . . . 6 ⊢ (𝑋 ∈ 𝐽 → ∀𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥) ∈ V)
47		dfiin2g 4962	. . . . . 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 481	. . . 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 2781	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧} = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
51	2	clsval 22188	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = ∩ {𝑧 ∈ (Clsd‘𝐽) ∣ 𝑆 ⊆ 𝑧})
52		uniiun 4988	. . . . . 6 ⊢ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) = ∪ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))𝑥
53	52	difeq2i 4054	. . . . 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 22053	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
56	55	adantr 481	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝐽)
57		0elpw 5278	. . . . . . 7 ⊢ ∅ ∈ 𝒫 (𝑋 ∖ 𝑆)
58	57	a1i 11	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ 𝒫 (𝑋 ∖ 𝑆))
59	56, 58	elind 4128	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)))
60		ne0i 4268	. . . . 5 ⊢ (∅ ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) → (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆)) ≠ ∅)
61		iindif2 5006	. . . . 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 2781	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))) = ∩ 𝑥 ∈ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))(𝑋 ∖ 𝑥))
64	50, 51, 63	3eqtr4d 2788	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
65		difssd 4067	. . . 4 ⊢ (𝑆 ⊆ 𝑋 → (𝑋 ∖ 𝑆) ⊆ 𝑋)
66	2	ntrval 22187	. . . 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 4057	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))) = (𝑋 ∖ ∪ (𝐽 ∩ 𝒫 (𝑋 ∖ 𝑆))))
69	64, 68	eqtr4d 2781	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ((cls‘𝐽)‘𝑆) = (𝑋 ∖ ((int‘𝐽)‘(𝑋 ∖ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  𝒫 cpw 4533  ∪ cuni 4839  ∩ cint 4879  ∪ ciun 4924  ∩ ciin 4925  ‘cfv 6433  Topctop 22042  Clsdccld 22167  intcnt 22168  clsccl 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170  df-ntr 22171  df-cls 22172

This theorem is referenced by:  ntrval2  22202  clsdif  22204  cmclsopn  22213  bcth3  24495