Theorem imasncls 23080

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 22303	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3	2	biimpi 215	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
4	3	ad2antlr 725	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5		imasnopn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	toptopon 22303	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
7	6	biimpi 215	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
8	7	ad2antrr 724	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9		simprr 771	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
10	4, 8, 9	cnmptc 23050	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
11	4	cnmptid 23049	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
12	4, 10, 11	cnmpt1t 23053	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13		simprl 769	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
14	5, 1	txuni 22980	. . . . 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 3987	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
17		eqid 2731	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
18	17	cncls2i 22658	. . 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 584	. 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 1917	. . . . 5 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋))
21		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
22		nfrab1 3424	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23		imass1 6058	. . . . . . . . . . 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 6142	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
26	25	ad2antll 727	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
27	24, 26	sseqtrd 3987	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
28	27	sseld 3946	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ 𝑌))
29	28	pm4.71rd 563	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
30		elimasng 6045	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
31	30	elvd 3453	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
32	31	ad2antll 727	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
33	32	anbi2d 629	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
34	29, 33	bitrd 278	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
35		rabid 3425	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
36	34, 35	bitr4di 288	. . . . 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 3966	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
38		eqid 2731	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩)
39	38	mptpreima 6195	. . . 4 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
40	37, 39	eqtr4di 2789	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
41	40	fveq2d 6851	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
42		nfcv 2902	. . . 4 ⊢ Ⅎ𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
43		nfrab1 3424	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
44		txtop 22957	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
45	44	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
46	17	clsss3 22447	. . . . . . . . . . . 12 ⊢ (((𝐽 ×t 𝐾) ∈ Top ∧ 𝑅 ⊆ ∪ (𝐽 ×t 𝐾)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
47	45, 16, 46	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ ∪ (𝐽 ×t 𝐾))
48	47, 15	sseqtrrd 3988	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
49		imass1 6058	. . . . . . . . . 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 3987	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
52	51	sseld 3946	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦 ∈ 𝑌))
53	52	pm4.71rd 563	. . . . . 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 6045	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
55	54	elvd 3453	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
56	55	ad2antll 727	. . . . . . 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 629	. . . . . 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 278	. . . . 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 3425	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
60	58, 59	bitr4di 288	. . . 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 3966	. . 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 6195	. . 3 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
63	61, 62	eqtr4di 2789	. 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 3994	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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3405  Vcvv 3446   ⊆ wss 3913  {csn 4591  ⟨cop 4597  ∪ cuni 4870   ↦ cmpt 5193   × cxp 5636  ^◡ccnv 5637   “ cima 5641  ‘cfv 6501  (class class class)co 7362  Topctop 22279  TopOnctopon 22296  clsccl 22406   Cn ccn 22612   ×_t ctx 22948

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-map 8774  df-topgen 17339  df-top 22280  df-topon 22297  df-bases 22333  df-cld 22407  df-cls 22409  df-cn 22615  df-cnp 22616  df-tx 22950

This theorem is referenced by:  utopreg  23641