Theorem cnpresti 23173

Description: One direction of cnprest 23174 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 2729	. . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnpf 23132	. . . 4 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐹:𝑋⟶∪ 𝐾)
4	3	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶∪ 𝐾)
5		simp1 1136	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ⊆ 𝑋)
6	4, 5	fssresd 6691	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴):𝐴⟶∪ 𝐾)
7		simpl2 1193	. . . . . 6 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → 𝑃 ∈ 𝐴)
8	7	fvresd 6842	. . . . 5 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → ((𝐹 ↾ 𝐴)‘𝑃) = (𝐹‘𝑃))
9	8	eleq1d 2813	. . . 4 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑦 ∈ 𝐾) → (((𝐹 ↾ 𝐴)‘𝑃) ∈ 𝑦 ↔ (𝐹‘𝑃) ∈ 𝑦))
10		cnpimaex 23141	. . . . . . 7 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾 ∧ (𝐹‘𝑃) ∈ 𝑦) → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦))
11	10	3expia 1121	. . . . . 6 ⊢ ((𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝐹 “ 𝑥) ⊆ 𝑦)))
12	11	3ad2antl3 1188	. . . . 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 1137	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝑃 ∈ 𝐴)
15	13, 14	jctird 526	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃 ∈ 𝑥 → (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴)))
16		elin 3919	. . . . . . . . . 10 ⊢ (𝑃 ∈ (𝑥 ∩ 𝐴) ↔ (𝑃 ∈ 𝑥 ∧ 𝑃 ∈ 𝐴))
17	15, 16	imbitrrdi 252	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑃 ∈ 𝑥 → 𝑃 ∈ (𝑥 ∩ 𝐴)))
18		inss1 4188	. . . . . . . . . . 11 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝑥
19		imass2 6053	. . . . . . . . . . 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 3947	. . . . . . . . 9 ⊢ ((𝐹 “ 𝑥) ⊆ 𝑦 → (𝐹 “ (𝑥 ∩ 𝐴)) ⊆ 𝑦)
23	17, 22	anim12d1 610	. . . . . . . 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 3144	. . . . . . 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 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
26	25	inex1 5256	. . . . . . . . 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 23127	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐽 ∈ Top)
29	28	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ Top)
30	29	uniexd 7678	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → ∪ 𝐽 ∈ V)
31	5, 1	sseqtrdi 3976	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ⊆ ∪ 𝐽)
32	30, 31	ssexd 5263	. . . . . . . . 9 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐴 ∈ V)
33		elrest 17331	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ V) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
34	29, 32, 33	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝑧 ∈ (𝐽 ↾t 𝐴) ↔ ∃𝑥 ∈ 𝐽 𝑧 = (𝑥 ∩ 𝐴)))
35		simpr 484	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → 𝑧 = (𝑥 ∩ 𝐴))
36	35	eleq2d 2814	. . . . . . . . 9 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → (𝑃 ∈ 𝑧 ↔ 𝑃 ∈ (𝑥 ∩ 𝐴)))
37	35	imaeq2d 6011	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = ((𝐹 ↾ 𝐴) “ (𝑥 ∩ 𝐴)))
38		inss2 4189	. . . . . . . . . . . 12 ⊢ (𝑥 ∩ 𝐴) ⊆ 𝐴
39		resima2 5967	. . . . . . . . . . . 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 2780	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) ∧ 𝑧 = (𝑥 ∩ 𝐴)) → ((𝐹 ↾ 𝐴) “ 𝑧) = (𝐹 “ (𝑥 ∩ 𝐴)))
42	41	sseq1d 3967	. . . . . . . . 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 632	. . . . . . . 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 5350	. . . . . . 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 259	. . . . . 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 480	. . . . 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 240	. . 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 3121	. 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 22802	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
51	29, 50	sylib 218	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐽 ∈ (TopOn‘𝑋))
52		resttopon 23046	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ⊆ 𝑋) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
53	51, 5, 52	syl2anc 584	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴))
54		cnptop2 23128	. . . . 5 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝐾 ∈ Top)
55	54	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ Top)
56	2	toptopon 22802	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
57	55, 56	sylib 218	. . 3 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐾 ∈ (TopOn‘∪ 𝐾))
58		iscnp 23122	. . 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 1373	. 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 713	1 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝑃 ∈ 𝐴 ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → (𝐹 ↾ 𝐴) ∈ (((𝐽 ↾t 𝐴) CnP 𝐾)‘𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ∩ cin 3902   ⊆ wss 3903  ∪ cuni 4858   ↾ cres 5621   “ cima 5622  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↾_t crest 17324  Topctop 22778  TopOnctopon 22795   CnP ccnp 23110

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-map 8755  df-en 8873  df-fin 8876  df-fi 9301  df-rest 17326  df-topgen 17347  df-top 22779  df-topon 22796  df-bases 22831  df-cnp 23113

This theorem is referenced by:  efrlim  26877  efrlimOLD  26878  cvmlift2lem11  35296