Theorem fcfnei 24064

Description: The property of being a cluster point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 26-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfnei	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))

Distinct variable groups:   𝐴,𝑛   𝑛,𝑠,𝐽   𝑛,𝐿,𝑠   𝑛,𝐹,𝑠   𝑛,𝑋,𝑠   𝑛,𝑌,𝑠

Allowed substitution hint:   𝐴(𝑠)

Proof of Theorem fcfnei

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfcf 24063	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))))
2		simpll1 1212	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3		topontop 22940	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
5		simpr 484	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ∈ ((nei‘𝐽)‘{𝐴}))
6		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	neii1 23135	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 583	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
9	6	ntrss2 23086	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
10	4, 8, 9	syl2anc 583	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ 𝑋)
12		toponuni 22941	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	2, 12	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑋 = ∪ 𝐽)
14	11, 13	eleqtrd 2846	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ∪ 𝐽)
15	14	snssd 4834	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
16	6	neiint 23133	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑛 ⊆ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
17	4, 15, 8, 16	syl3anc 1371	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
18	5, 17	mpbid 232	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑛))
19		snssg 4808	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑛) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
20	11, 19	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (𝐴 ∈ ((int‘𝐽)‘𝑛) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
21	18, 20	mpbird 257	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑛))
22	6	ntropn 23078	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
23	4, 8, 22	syl2anc 583	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
24		eleq2 2833	. . . . . . . . . 10 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ ((int‘𝐽)‘𝑛)))
25		ineq1 4234	. . . . . . . . . . . 12 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (𝑜 ∩ (𝐹 “ 𝑠)) = (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)))
26	25	neeq1d 3006	. . . . . . . . . . 11 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → ((𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅))
27	26	ralbidv 3184	. . . . . . . . . 10 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅))
28	24, 27	imbi12d 344	. . . . . . . . 9 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → ((𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) ↔ (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅)))
29	28	rspcv 3631	. . . . . . . 8 ⊢ (((int‘𝐽)‘𝑛) ∈ 𝐽 → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅)))
30	23, 29	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝐴 ∈ ((int‘𝐽)‘𝑛) → ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅)))
31	21, 30	mpid 44	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅))
32		ssrin 4263	. . . . . . . 8 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)))
33		ssn0 4427	. . . . . . . . 9 ⊢ (((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)) ∧ (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)
34	33	ex 412	. . . . . . . 8 ⊢ ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)) → ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
35	32, 34	syl 17	. . . . . . 7 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
36	35	ralimdv 3175	. . . . . 6 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
37	10, 31, 36	sylsyld 61	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
38	37	ralrimdva 3160	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
39		simpl1 1191	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
40	39, 3	syl 17	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
41		opnneip 23148	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42	41	3expb 1120	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
43	40, 42	sylan 579	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44		ineq1 4234	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑜 ∩ (𝐹 “ 𝑠)))
45	44	neeq1d 3006	. . . . . . . . . 10 ⊢ (𝑛 = 𝑜 → ((𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
46	45	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑛 = 𝑜 → (∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
47	46	rspcv 3631	. . . . . . . 8 ⊢ (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
48	43, 47	syl 17	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
49	48	expr 456	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (𝐴 ∈ 𝑜 → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
50	49	com23 86	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
51	50	ralrimdva 3160	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
52	38, 51	impbid 212	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
53	52	pm5.32da 578	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))
54	1, 53	bitrd 279	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∀wral 3067   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  {csn 4648  ∪ cuni 4931   “ cima 5703  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937  intcnt 23046  neicnei 23126  Filcfil 23874   fClusf cfcf 23966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-fbas 21384  df-fg 21385  df-top 22921  df-topon 22938  df-cld 23048  df-ntr 23049  df-cls 23050  df-nei 23127  df-fil 23875  df-fm 23967  df-fcls 23970  df-fcf 23971

This theorem is referenced by:  fcfneii  24066