Theorem flimrest 24023

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 1148	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23892	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1142	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23201	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
5	1, 3, 4	syl2anc 593	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐽 ↾t 𝑌) ∈ (TopOn‘𝑌))
6		filfbas 23888	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1146	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1150	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23879	. . . . . . 7 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
10	7, 8, 9	syl2anc 593	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ¬ (𝑋 ∖ 𝑌) ∈ 𝐹)
11		simp2 1149	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23928	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
13	11, 3, 12	syl2anc 593	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐹 ↾t 𝑌) ∈ (Fil‘𝑌) ↔ ¬ (𝑋 ∖ 𝑌) ∈ 𝐹))
14	10, 13	mpbird 259	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾t 𝑌) ∈ (Fil‘𝑌))
15		flimopn 24015	. . . . 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 593	. . . 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 1226	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝐹 ∈ (Fil‘𝑋))
18		simpll3 1227	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑌 ∈ 𝐹)
19		elrestr 17440	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))
20	19	3expia 1133	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
21	17, 18, 20	syl2anc 593	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∈ 𝐹 → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
22		trfilss 23929	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
23	17, 18, 22	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝐹 ↾t 𝑌) ⊆ 𝐹)
24	23	sseld 3935	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌) → (𝑧 ∩ 𝑌) ∈ 𝐹))
25		inss1 4188	. . . . . . . . . . . 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 1204	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28		toponss 22967	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
29	27, 28	sylan 589	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑧 ⊆ 𝑋)
30		filss 23893	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧 ∩ 𝑌) ∈ 𝐹 ∧ 𝑧 ⊆ 𝑋 ∧ (𝑧 ∩ 𝑌) ⊆ 𝑧)) → 𝑧 ∈ 𝐹)
31	30	3exp2 1367	. . . . . . . . . . . 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 214	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → (𝑧 ∈ 𝐹 ↔ (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
36	35	imbi2d 342	. . . . . . 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 3182	. . . . . 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 1205	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
39	3	sselda 3936	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
40		flimopn 24015	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹))))
41	40	baibd 547	. . . . . . 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 848	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹)))
43		vex 3457	. . . . . . . . 9 ⊢ 𝑧 ∈ V
44	43	inex1 5272	. . . . . . . 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 1206	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐹)
47		elrest 17439	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
48	27, 46, 47	syl2anc 593	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑦 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑧 ∈ 𝐽 𝑦 = (𝑧 ∩ 𝑌)))
49		eleq2 2850	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝑌)))
50		elin 3920	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ (𝑥 ∈ 𝑧 ∧ 𝑥 ∈ 𝑌))
51	50	rbaib 546	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ 𝑥 ∈ 𝑧))
52	51	adantl 485	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝑧 ∩ 𝑌) ↔ 𝑥 ∈ 𝑧))
53	49, 52	sylan9bbr 518	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑦 = (𝑧 ∩ 𝑌)) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ 𝑧))
54		eleq1 2849	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑦 ∈ (𝐹 ↾t 𝑌) ↔ (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌)))
55	54	adantl 485	. . . . . . . 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 346	. . . . . . 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 5366	. . . . . 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 313	. . . . 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 587	. . . 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 284	. . 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 464	. . . 4 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ 𝑌))
62		elin 3920	. . . 4 ⊢ (𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝐹) ∧ 𝑥 ∈ 𝑌))
63	61, 62	bitr4i 280	. . 3 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑥 ∈ (𝐽 fLim 𝐹)) ↔ 𝑥 ∈ ((𝐽 fLim 𝐹) ∩ 𝑌))
64	60, 63	bitrdi 289	. 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 2759	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  Vcvv 3453   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ‘cfv 6517  (class class class)co 7392   ↾_t crest 17432  fBascfbas 21392  TopOnctopon 22950  Filcfil 23885   fLim cflim 23974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-en 8924  df-fin 8927  df-fi 9354  df-rest 17434  df-topgen 17455  df-fbas 21401  df-fg 21402  df-top 22934  df-topon 22951  df-bases 22986  df-ntr 23060  df-nei 23138  df-fil 23886  df-flim 23979

This theorem is referenced by:  metsscmetcld  25357  cmetss  25358  minveclem4a  25472