Theorem flimrest 23958

Description: The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.)

Assertion

Ref	Expression
flimrest	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))

Proof of Theorem flimrest

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

Step	Hyp	Ref	Expression
1		simp1 1137	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23827	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1131	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23136	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
5	1, 3, 4	syl2anc 585	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
6		filfbas 23823	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1139	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23814	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
10	7, 8, 9	syl2anc 585	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
11		simp2 1138	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23863	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
13	11, 3, 12	syl2anc 585	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐹 ↾t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
14	10, 13	mpbird 257	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾t 𝑌) ∈ (Fil‘𝑌))
15		flimopn 23950	. . . . 5 ⊢ (((𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌) ∧ (𝐹 ↾t 𝑌) ∈ (Fil‘𝑌)) → (𝑥 ∈ ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾t 𝑌)(𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌)))))
16	5, 14, 15	syl2anc 585	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾t 𝑌)(𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌)))))
17		simpll2 1215	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝐹 ∈ (Fil‘𝑋))
18		simpll3 1216	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑌 ∈ 𝐹)
19		elrestr 17382	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))
20	19	3expia 1122	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
21	17, 18, 20	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
22		trfilss 23864	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
23	17, 18, 22	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
24	23	sseld 3921	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌) → (𝑧 ∩ 𝑌) ∈ 𝐹))
25		inss1 4178	. . . . . . . . . . . 12 ⊢ (𝑧 ∩ 𝑌) ⊆ 𝑧
26	25	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑌) ⊆ 𝑧)
27		simpl1 1193	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28		toponss 22902	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
29	27, 28	sylan 581	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
30		filss 23828	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧 ∩ 𝑌) ∈ 𝐹 ∧ 𝑧 ⊆ 𝑋 ∧ (𝑧 ∩ 𝑌) ⊆ 𝑧)) → 𝑧 ∈ 𝐹)
31	30	3exp2 1356	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑧 ∩ 𝑌) ∈ 𝐹 → (𝑧 ⊆ 𝑋 → ((𝑧 ∩ 𝑌) ⊆ 𝑧 → 𝑧 ∈ 𝐹))))
32	31	com24 95	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑧 ∩ 𝑌) ⊆ 𝑧 → (𝑧 ⊆ 𝑋 → ((𝑧 ∩ 𝑌) ∈ 𝐹 → 𝑧 ∈ 𝐹))))
33	17, 26, 29, 32	syl3c 66	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ 𝐹 → 𝑧 ∈ 𝐹))
34	24, 33	syld 47	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌) → 𝑧 ∈ 𝐹))
35	21, 34	impbid 212	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∈ 𝐹 ↔ (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
36	35	imbi2d 340	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹) ↔ (𝑥 ∈ 𝑧 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))))
37	36	ralbidva 3159	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))))
38		simpl2 1194	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
39	3	sselda 3922	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
40		flimopn 23950	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹))))
41	40	baibd 539	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹)))
42	27, 38, 39, 41	syl21anc 838	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹)))
43		vex 3434	. . . . . . . . 9 ⊢ 𝑧 ∈ V
44	43	inex1 5254	. . . . . . . 8 ⊢ (𝑧 ∩ 𝑌) ∈ V
45	44	a1i 11	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∩ 𝑌) ∈ V)
46		simpl3 1195	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐹)
47		elrest 17381	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
48	27, 46, 47	syl2anc 585	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
49		eleq2 2826	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝑌)))
50		elin 3906	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑌))
51	50	rbaib 538	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ 𝑥 ∈ 𝑧))
52	51	adantl 481	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ 𝑥 ∈ 𝑧))
53	49, 52	sylan9bbr 510	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑧 ∩ 𝑌)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
54		eleq1 2825	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑦 ∈ (𝐹 ↾t 𝑌) ↔ (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
55	54	adantl 481	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑧 ∩ 𝑌)) → (𝑦 ∈ (𝐹 ↾t 𝑌) ↔ (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
56	53, 55	imbi12d 344	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑧 ∩ 𝑌)) → ((𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌)) ↔ (𝑥 ∈ 𝑧 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))))
57	45, 48, 56	ralxfr2d 5347	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (∀𝑦 ∈ (𝐽 ↾t 𝑌)(𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌)) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))))
58	37, 42, 57	3bitr4d 311	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑦 ∈ (𝐽 ↾t 𝑌)(𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌))))
59	58	pm5.32da 579	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ 𝑌 ∧ ∀𝑦 ∈ (𝐽 ↾t 𝑌)(𝑥 ∈ 𝑦 → 𝑦 ∈ (𝐹 ↾t 𝑌)))))
60	16, 59	bitr4d 282	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) ↔ (𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹))))
61		ancom 460	. . . 4 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ 𝑌))
62		elin 3906	. . . 4 ⊢ (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ 𝑌))
63	61, 62	bitr4i 278	. . 3 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
64	60, 63	bitrdi 287	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑥 ∈ ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌)))
65	64	eqrdv 2735	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7360   ↾_t crest 17374  fBascfbas 21332  TopOnctopon 22885  Filcfil 23820   fLim cflim 23909

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-en 8887  df-fin 8890  df-fi 9317  df-rest 17376  df-topgen 17397  df-fbas 21341  df-fg 21342  df-top 22869  df-topon 22886  df-bases 22921  df-ntr 22995  df-nei 23073  df-fil 23821  df-flim 23914

This theorem is referenced by:  metsscmetcld  25292  cmetss  25293  minveclem4a  25407