Theorem flimrest 23927

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 1136	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐽 ∈ (TopOn‘𝑋))
2		filelss 23796	. . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
3	2	3adant1 1130	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ⊆ 𝑋)
4		resttopon 23105	. . . . . 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 23792	. . . . . . . 8 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
7	6	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (fBas‘𝑋))
8		simp3 1138	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝑌 ∈ 𝐹)
9		fbncp 23783	. . . . . . 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 1137	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → 𝐹 ∈ (Fil‘𝑋))
12		trfil3 23832	. . . . . . 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 23919	. . . . 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 1214	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝐹 ∈ (Fil‘𝑋))
18		simpll3 1215	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → 𝑌 ∈ 𝐹)
19		elrestr 17348	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹 ∧ 𝑧 ∈ 𝐹) → (𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌))
20	19	3expia 1121	. . . . . . . . . 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 23833	. . . . . . . . . . . 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 3932	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) ∧ 𝑧 ∈ 𝐽) → ((𝑧 ∩ 𝑌) ∈ (𝐹 ↾t 𝑌) → (𝑧 ∩ 𝑌) ∈ 𝐹))
25		inss1 4189	. . . . . . . . . . . 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 1192	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐽 ∈ (TopOn‘𝑋))
28		toponss 22871	. . . . . . . . . . . 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 23797	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ ((𝑧 ∩ 𝑌) ∈ 𝐹 ∧ 𝑧 ⊆ 𝑋 ∧ (𝑧 ∩ 𝑌) ⊆ 𝑧)) → 𝑧 ∈ 𝐹)
31	30	3exp2 1355	. . . . . . . . . . . 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 3157	. . . . . 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 1193	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝐹 ∈ (Fil‘𝑋))
39	3	sselda 3933	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑋)
40		flimopn 23919	. . . . . . . 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 837	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ ∀𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 → 𝑧 ∈ 𝐹)))
43		vex 3444	. . . . . . . . 9 ⊢ 𝑧 ∈ V
44	43	inex1 5262	. . . . . . . 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 1194	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐹)
47		elrest 17347	. . . . . . . 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 2825	. . . . . . . . 9 ⊢ (𝑦 = (𝑧 ∩ 𝑌) → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑧 ∩ 𝑌)))
50		elin 3917	. . . . . . . . . . 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 2824	. . . . . . . . 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 5355	. . . . . 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 3917	. . . 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 2734	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 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358   ↾_t crest 17340  fBascfbas 21297  TopOnctopon 22854  Filcfil 23789   fLim cflim 23878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-en 8884  df-fin 8887  df-fi 9314  df-rest 17342  df-topgen 17363  df-fbas 21306  df-fg 21307  df-top 22838  df-topon 22855  df-bases 22890  df-ntr 22964  df-nei 23042  df-fil 23790  df-flim 23883

This theorem is referenced by:  metsscmetcld  25271  cmetss  25272  minveclem4a  25386