Theorem imasncls 21495

Description: If a relation graph is closed, then an image set of a singleton is also closed. Corollary of proposition 4 of [BourbakiTop1] p. I.26. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Hypotheses

Ref	Expression
imasnopn.1	⊢ 𝑋 = ∪ 𝐽
imasnopn.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
imasncls	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))

Proof of Theorem imasncls

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasnopn.2	. . . . . . 7 ⊢ 𝑌 = ∪ 𝐾
2	1	toptopon 20722	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3	2	biimpi 206	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
4	3	ad2antlr 763	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5		imasnopn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	toptopon 20722	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
7	6	biimpi 206	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
8	7	ad2antrr 762	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9		simprr 796	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
10	4, 8, 9	cnmptc 21465	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
11	4	cnmptid 21464	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
12	4, 10, 11	cnmpt1t 21468	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13		simprl 794	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
14	5, 1	txuni 21395	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
15	14	adantr 481	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
16	13, 15	sseqtrd 3641	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
17		eqid 2622	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
18	17	cncls2i 21074	. . 3 ⊢ (((𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)) ∧ 𝑅 ⊆ ∪ (𝐽 ×t 𝐾)) → ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)) ⊆ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
19	12, 16, 18	syl2anc 693	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)) ⊆ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
20		nfv 1843	. . . . 5 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋))
21		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
22		nfrab1 3122	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23		imass1 5500	. . . . . . . . . . 11 ⊢ (𝑅 ⊆ (𝑋 × 𝑌) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
24	13, 23	syl 17	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
25		xpimasn 5579	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
26	25	ad2antll 765	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
27	24, 26	sseqtrd 3641	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
28	27	sseld 3602	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ 𝑌))
29	28	pm4.71rd 667	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
30		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
31		elimasng 5491	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
32	30, 31	mpan2 707	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
33	32	ad2antll 765	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
34	33	anbi2d 740	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
35	29, 34	bitrd 268	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
36		rabid 3116	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
37	35, 36	syl6bbr 278	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
38	20, 21, 22, 37	eqrd 3622	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
39		eqid 2622	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩)
40	39	mptpreima 5628	. . . 4 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
41	38, 40	syl6eqr 2674	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
42	41	fveq2d 6195	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
43		nfcv 2764	. . . 4 ⊢ Ⅎ𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
44		nfrab1 3122	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
45		txtop 21372	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
46	45	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
47	17	clsss3 20863	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ⊆ ∪ (𝐽 ×t 𝐾)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
48	46, 16, 47	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
49	48, 15	sseqtr4d 3642	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
50		imass1 5500	. . . . . . . . . 10 ⊢ (((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
51	49, 50	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
52	51, 26	sseqtrd 3641	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
53	52	sseld 3602	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦 ∈ 𝑌))
54	53	pm4.71rd 667	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))))
55		elimasng 5491	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
56	30, 55	mpan2 707	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
57	56	ad2antll 765	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
58	57	anbi2d 740	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
59	54, 58	bitrd 268	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅))))
60		rabid 3116	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
61	59, 60	syl6bbr 278	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ 𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}))
62	20, 43, 44, 61	eqrd 3622	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)})
63	39	mptpreima 5628	. . 3 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
64	62, 63	syl6eqr 2674	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) = (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
65	19, 42, 64	3sstr4d 3648	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) ⊆ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ∪ cuni 4436   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   “ cima 5117  ‘cfv 5888  (class class class)co 6650  Topctop 20698  TopOnctopon 20715  clsccl 20822   Cn ccn 21028   ×_t ctx 21363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-cls 20825  df-cn 21031  df-cnp 21032  df-tx 21365

This theorem is referenced by:  utopreg  22056