Theorem imasncls 23555

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 22780	. . . . . 6 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
3	2	biimpi 216	. . . . 5 ⊢ (𝐾 ∈ Top → 𝐾 ∈ (TopOn‘𝑌))
4	3	ad2antlr 727	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
5		imasnopn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
6	5	toptopon 22780	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
7	6	biimpi 216	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝑋))
8	7	ad2antrr 726	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
9		simprr 772	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
10	4, 8, 9	cnmptc 23525	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝐴) ∈ (𝐾 Cn 𝐽))
11	4	cnmptid 23524	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ 𝑦) ∈ (𝐾 Cn 𝐾))
12	4, 10, 11	cnmpt1t 23528	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) ∈ (𝐾 Cn (𝐽 ×t 𝐾)))
13		simprl 770	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ (𝑋 × 𝑌))
14	5, 1	txuni 23455	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
15	14	adantr 480	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑋 × 𝑌) = ∪ (𝐽 ×t 𝐾))
16	13, 15	sseqtrd 3980	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → 𝑅 ⊆ ∪ (𝐽 ×t 𝐾))
17		eqid 2729	. . . 4 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
18	17	cncls2i 23133	. . 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 1914	. . . . 5 ⊢ Ⅎ𝑦((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋))
21		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦(𝑅 “ {𝐴})
22		nfrab1 3423	. . . . 5 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
23		imass1 6061	. . . . . . . . . . 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 6146	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑋 → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
26	25	ad2antll 729	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑋 × 𝑌) “ {𝐴}) = 𝑌)
27	24, 26	sseqtrd 3980	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) ⊆ 𝑌)
28	27	sseld 3942	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) → 𝑦 ∈ 𝑌))
29	28	pm4.71rd 562	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴}))))
30		elimasng 6049	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
31	30	elvd 3450	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
32	31	ad2antll 729	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
33	32	anbi2d 630	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((𝑦 ∈ 𝑌 ∧ 𝑦 ∈ (𝑅 “ {𝐴})) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
34	29, 33	bitrd 279	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (𝑅 “ {𝐴}) ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅)))
35		rabid 3424	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ 𝑅))
36	34, 35	bitr4di 289	. . . . 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 3963	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅})
38		eqid 2729	. . . . 5 ⊢ (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) = (𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩)
39	38	mptpreima 6199	. . . 4 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ 𝑅}
40	37, 39	eqtr4di 2782	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑅 “ {𝐴}) = (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅))
41	40	fveq2d 6844	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘𝐾)‘(𝑅 “ {𝐴})) = ((cls‘𝐾)‘(◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ 𝑅)))
42		nfcv 2891	. . . 4 ⊢ Ⅎ𝑦(((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴})
43		nfrab1 3423	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
44		txtop 23432	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 ×t 𝐾) ∈ Top)
45	44	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝐽 ×t 𝐾) ∈ Top)
46	17	clsss3 22922	. . . . . . . . . . . 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 3981	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → ((cls‘(𝐽 ×t 𝐾))‘𝑅) ⊆ (𝑋 × 𝑌))
49		imass1 6061	. . . . . . . . . 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 3980	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ⊆ 𝑌)
52	51	sseld 3942	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝑅 ⊆ (𝑋 × 𝑌) ∧ 𝐴 ∈ 𝑋)) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) → 𝑦 ∈ 𝑌))
53	52	pm4.71rd 562	. . . . . 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 6049	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝑦 ∈ V) → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
55	54	elvd 3450	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑋 → (𝑦 ∈ (((cls‘(𝐽 ×t 𝐾))‘𝑅) “ {𝐴}) ↔ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
56	55	ad2antll 729	. . . . . . 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 630	. . . . . 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 279	. . . . 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 3424	. . . . 5 ⊢ (𝑦 ∈ {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)} ↔ (𝑦 ∈ 𝑌 ∧ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)))
60	58, 59	bitr4di 289	. . . 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 3963	. . 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 6199	. . 3 ⊢ (◡(𝑦 ∈ 𝑌 ↦ ⟨𝐴, 𝑦⟩) “ ((cls‘(𝐽 ×t 𝐾))‘𝑅)) = {𝑦 ∈ 𝑌 ∣ ⟨𝐴, 𝑦⟩ ∈ ((cls‘(𝐽 ×t 𝐾))‘𝑅)}
63	61, 62	eqtr4di 2782	. 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 3999	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 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3402  Vcvv 3444   ⊆ wss 3911  {csn 4585  ⟨cop 4591  ∪ cuni 4867   ↦ cmpt 5183   × cxp 5629  ^◡ccnv 5630   “ cima 5634  ‘cfv 6499  (class class class)co 7369  Topctop 22756  TopOnctopon 22773  clsccl 22881   Cn ccn 23087   ×_t ctx 23423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-cld 22882  df-cls 22884  df-cn 23090  df-cnp 23091  df-tx 23425

This theorem is referenced by:  utopreg  24116