Theorem flimrest 24006

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 1135	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23875	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1129	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23184	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
5	1, 3, 4	syl2anc 584	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
6		filfbas 23871	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1133	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1137	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23862	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
10	7, 8, 9	syl2anc 584	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
11		simp2 1136	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23911	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
13	11, 3, 12	syl2anc 584	. . . . . 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 23998	. . . . 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 584	. . . 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 1212	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝐹 ∈ (Fil‘𝑋))
18		simpll3 1213	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑌 ∈ 𝐹)
19		elrestr 17474	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))
20	19	3expia 1120	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
21	17, 18, 20	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
22		trfilss 23912	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
23	17, 18, 22	syl2anc 584	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
24	23	sseld 3993	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌) → (𝑧 ∩ 𝑌) ∈ 𝐹))
25		inss1 4244	. . . . . . . . . . . 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 1190	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28		toponss 22948	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
29	27, 28	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
30		filss 23876	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧 ∩ 𝑌) ∈ 𝐹 ∧ 𝑧 ⊆ 𝑋 ∧ (𝑧 ∩ 𝑌) ⊆ 𝑧)) → 𝑧 ∈ 𝐹)
31	30	3exp2 1353	. . . . . . . . . . . 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 3173	. . . . . 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 1191	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
39	3	sselda 3994	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
40		flimopn 23998	. . . . . . . 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 3481	. . . . . . . . 9 ⊢ 𝑧 ∈ V
44	43	inex1 5322	. . . . . . . 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 1192	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐹)
47		elrest 17473	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
48	27, 46, 47	syl2anc 584	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
49		eleq2 2827	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝑌)))
50		elin 3978	. . . . . . . . . . 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 2826	. . . . . . . . 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 5415	. . . . . 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 3978	. . . 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 2732	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 1086   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ∖ cdif 3959   ∩ cin 3961   ⊆ wss 3962  ‘cfv 6562  (class class class)co 7430   ↾_t crest 17466  fBascfbas 21369  TopOnctopon 22931  Filcfil 23868   fLim cflim 23957

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-en 8984  df-fin 8987  df-fi 9448  df-rest 17468  df-topgen 17489  df-fbas 21378  df-fg 21379  df-top 22915  df-topon 22932  df-bases 22968  df-ntr 23043  df-nei 23121  df-fil 23869  df-flim 23962

This theorem is referenced by:  metsscmetcld  25362  cmetss  25363  minveclem4a  25477