Theorem imasncls 23752

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 22977	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3	2	biimpi 218	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
4	3	ad2antlr 737	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5		imasnopn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	toptopon 22977	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
7	6	biimpi 218	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
8	7	ad2antrr 736	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9		simprr 782	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
10	4, 8, 9	cnmptc 23722	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
11	4	cnmptid 23721	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
12	4, 10, 11	cnmpt1t 23725	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13		simprl 780	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
14	5, 1	txuni 23652	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
15	14	adantr 484	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
16	13, 15	sseqtrd 3972	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
17		eqid 2762	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
18	17	cncls2i 23330	. . 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 593	. 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 1934	. . . . 5 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋))
21		nfcv 2924	. . . . 5 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
22		nfrab1 3434	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23		imass1 6090	. . . . . . . . . . 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 6171	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
26	25	ad2antll 739	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
27	24, 26	sseqtrd 3972	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
28	27	sseld 3935	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ 𝑌))
29	28	pm4.71rd 570	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
30		elimasng 6078	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
31	30	elvd 3460	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
32	31	ad2antll 739	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
33	32	anbi2d 639	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
34	29, 33	bitrd 281	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
35		rabid 3435	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
36	34, 35	bitr4di 291	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ 𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}))
37	20, 21, 22, 36	eqrd 3955	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
38		eqid 2762	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩)
39	38	mptpreima 6225	. . . 4 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
40	37, 39	eqtr4di 2815	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
41	40	fveq2d 6871	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
42		nfcv 2924	. . . 4 ⊢ Ⅎ𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
43		nfrab1 3434	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
44		txtop 23629	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
45	44	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
46	17	clsss3 23119	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ⊆ ∪ (𝐽 ×t 𝐾)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
47	45, 16, 46	syl2anc 593	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
48	47, 15	sseqtrrd 3973	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
49		imass1 6090	. . . . . . . . . 10 ⊢ (((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
50	48, 49	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ ((𝑋 × 𝑌) “ {𝐴}))
51	50, 26	sseqtrd 3972	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
52	51	sseld 3935	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦 ∈ 𝑌))
53	52	pm4.71rd 570	. . . . . 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}))))
54		elimasng 6078	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
55	54	elvd 3460	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
56	55	ad2antll 739	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
57	56	anbi2d 639	. . . . . 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))))
58	53, 57	bitrd 281	. . . . 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))))
59		rabid 3435	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
60	58, 59	bitr4di 291	. . . 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)}))
61	20, 42, 43, 60	eqrd 3955	. . 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)})
62	38	mptpreima 6225	. . 3 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
63	61, 62	eqtr4di 2815	. 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)))
64	19, 41, 63	3sstr4d 3991	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 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  {crab 3414  Vcvv 3454   ⊆ wss 3904  {csn 4582  ⟨cop 4588  ∪ cuni 4865   ↦ cmpt 5181   × cxp 5645  ^◡ccnv 5646   “ cima 5650  ‘cfv 6521  (class class class)co 7396  Topctop 22953  TopOnctopon 22970  clsccl 23078   Cn ccn 23284   ×_t ctx 23620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-map 8810  df-topgen 17472  df-top 22954  df-topon 22971  df-bases 23006  df-cld 23079  df-cls 23081  df-cn 23287  df-cnp 23288  df-tx 23622

This theorem is referenced by:  utopreg  24312