Theorem fcfnei 24083

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 24082	. 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 1225	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3		topontop 22961	. . . . . . . 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 488	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ∈ ((nei‘𝐽)‘{𝐴}))
6		eqid 2761	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	neii1 23154	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 593	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
9	6	ntrss2 23105	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
10	4, 8, 9	syl2anc 593	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11		simplr 778	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ 𝑋)
12		toponuni 22962	. . . . . . . . . . . . 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 2863	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ∪ 𝐽)
15	14	snssd 4742	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
16	6	neiint 23152	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑛 ⊆ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
17	4, 15, 8, 16	syl3anc 1389	. . . . . . . . 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 234	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑛))
19		snssg 4739	. . . . . . . . 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 259	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑛))
22	6	ntropn 23097	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
23	4, 8, 22	syl2anc 593	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
24		eleq2 2850	. . . . . . . . . 10 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (𝐴 ∈ 𝑜 ↔ 𝐴 ∈ ((int‘𝐽)‘𝑛)))
25		ineq1 4163	. . . . . . . . . . . 12 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (𝑜 ∩ (𝐹 “ 𝑠)) = (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)))
26	25	neeq1d 3015	. . . . . . . . . . 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 346	. . . . . . . . 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 3576	. . . . . . . 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 4191	. . . . . . . 8 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)))
33		ssn0 4355	. . . . . . . . 9 ⊢ (((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)) ∧ (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅)
34	33	ex 416	. . . . . . . 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 3161	. . . 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 1204	. . . . . . . . . 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 23167	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42	41	3expb 1132	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
43	40, 42	sylan 589	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44		ineq1 4163	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑜 → (𝑛 ∩ (𝐹 “ 𝑠)) = (𝑜 ∩ (𝐹 “ 𝑠)))
45	44	neeq1d 3015	. . . . . . . . . 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 3576	. . . . . . . 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 460	. . . . . 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 3161	. . . 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 214	. . 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 587	. 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 281	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075   ∩ cin 3901   ⊆ wss 3902  ∅c0 4283  {csn 4579  ∪ cuni 4862   “ cima 5646  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391  Topctop 22941  TopOnctopon 22958  intcnt 23065  neicnei 23145  Filcfil 23893   fClusf cfcf 23985

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 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  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 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-iin 4949  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-map 8804  df-fbas 21409  df-fg 21410  df-top 22942  df-topon 22959  df-cld 23067  df-ntr 23068  df-cls 23069  df-nei 23146  df-fil 23894  df-fm 23986  df-fcls 23989  df-fcf 23990

This theorem is referenced by:  fcfneii  24085