Theorem cnpresti 21893

Description: One direction of cnprest 21894 under the weaker condition that the point is in the subset rather than the interior of the subset. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 1-May-2015.)

Hypothesis

Ref	Expression
cnprest.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cnpresti	⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))

Proof of Theorem cnpresti

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnprest.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2		eqid 2798	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 21852	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶∪ 𝐾)
4	3	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶∪ 𝐾)
5		simp1 1133	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ⊆ 𝑋)
6	4, 5	fssresd 6519	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾)
7		simpl2 1189	. . . . . 6 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → 𝑃 ∈ 𝐴)
8	7	fvresd 6665	. . . . 5 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
9	8	eleq1d 2874	. . . 4 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹‘𝑃) ∈ 𝑦))
10		cnpimaex 21861	. . . . . . 7 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
11	10	3expia 1118	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
12	11	3ad2antl3 1184	. . . . 5 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
13		idd 24	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃 ∈ 𝑥 → 𝑃 ∈ 𝑥))
14		simp2 1134	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝐴)
15	13, 14	jctird 530	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃 ∈ 𝑥 → (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴)))
16		elin 3897	. . . . . . . . . 10 ⊢ (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴))
17	15, 16	syl6ibr 255	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃 ∈ 𝑥 → 𝑃 ∈ (𝑥 ∩ 𝐴)))
18		inss1 4155	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
19		imass2 5932	. . . . . . . . . . 11 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝑥 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥))
20	18, 19	ax-mp 5	. . . . . . . . . 10 ⊢ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ (𝐹 “ 𝑥)
21		id 22	. . . . . . . . . 10 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ 𝑥) ⊆ 𝑦)
22	20, 21	sstrid 3926	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)
23	17, 22	anim12d1 612	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
24	23	reximdv 3232	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
25		vex 3444	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
26	25	inex1 5185	. . . . . . . . 9 ⊢ (𝑥 ∩ 𝐴) ∈ V
27	26	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∩ 𝐴) ∈ V)
28		cnptop1 21847	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
29	28	3ad2ant3 1132	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
30	29	uniexd 7448	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∪ 𝐽 ∈ V)
31	5, 1	sseqtrdi 3965	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ⊆ ∪ 𝐽)
32	30, 31	ssexd 5192	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
33		elrest 16693	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
34	29, 32, 33	syl2anc 587	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
35		simpr 488	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → 𝑧 = (𝑥 ∩ 𝐴))
36	35	eleq2d 2875	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (𝑥 ∩ 𝐴)))
37	35	imaeq2d 5896	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)))
38		inss2 4156	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
39		resima2 5853	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ 𝐴) ⊆ 𝐴 → ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴)))
40	38, 39	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)) = (𝐹 “ (𝑥 ∩ 𝐴))
41	37, 40	eqtrdi 2849	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = (𝐹 “ (𝑥 ∩ 𝐴)))
42	41	sseq1d 3946	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦 ↔ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦))
43	36, 42	anbi12d 633	. . . . . . . 8 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
44	27, 34, 43	rexxfr2d 5277	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ (𝑥 ∩ 𝐴) ∧ (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)))
45	24, 44	sylibrd 262	. . . . . 6 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
46	45	adantr 484	. . . . 5 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → (∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦) → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
47	12, 46	syld 47	. . . 4 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
48	9, 47	sylbid 243	. . 3 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
49	48	ralrimiva 3149	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))
50	1	toptopon 21522	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
51	29, 50	sylib 221	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
52		resttopon 21766	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
53	51, 5, 52	syl2anc 587	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
54		cnptop2 21848	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
55	54	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
56	2	toptopon 21522	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
57	55, 56	sylib 221	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
58		iscnp 21842	. . 3 ⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝑃 ∈ 𝐴) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
59	53, 57, 14, 58	syl3anc 1368	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ((𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃) ↔ ((𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 → ∃𝑧 ∈ (𝐽 ↾t 𝐴)(𝑃 ∈ 𝑧 ∧ ((𝐹 ↾ 𝐴) “ 𝑧) ⊆ 𝑦)))))
60	6, 49, 59	mpbir2and 712	1 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  ∪ cuni 4800   ↾ cres 5521   “ cima 5522  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↾_t crest 16686  Topctop 21498  TopOnctopon 21515   CnP ccnp 21830

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16688  df-topgen 16709  df-top 21499  df-topon 21516  df-bases 21551  df-cnp 21833

This theorem is referenced by:  efrlim  25555  cvmlift2lem11  32673