Theorem fcfnei 24347

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 24346	. 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 1231	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ (TopOn‘𝑋))
3		topontop 23224	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
4	2, 3	syl 18	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐽 ∈ Top)
5		simpr 490	. . . . . . . 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 23417	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
8	4, 5, 7	syl2anc 596	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑛 ⊆ ∪ 𝐽)
9	6	ntrss2 23368	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
10	4, 8, 9	syl2anc 596	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → ((int‘𝐽)‘𝑛) ⊆ 𝑛)
11		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ 𝑋)
12		toponuni 23225	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	2, 12	syl 18	. . . . . . . . . . . 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 4747	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ∪ 𝐽)
16	6	neiint 23415	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ {𝐴} ⊆ ∪ 𝐽 ∧ 𝑛 ⊆ ∪ 𝐽) → (𝑛 ∈ ((nei‘𝐽)‘{𝐴}) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
17	4, 15, 8, 16	syl3anc 1398	. . . . . . . . 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 235	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → {𝐴} ⊆ ((int‘𝐽)‘𝑛))
19		snssg 4744	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∈ ((int‘𝐽)‘𝑛) ↔ {𝐴} ⊆ ((int‘𝐽)‘𝑛)))
20	11, 19	syl 18	. . . . . . . 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 260	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝐴})) → 𝐴 ∈ ((int‘𝐽)‘𝑛))
22	6	ntropn 23360	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑛 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑛) ∈ 𝐽)
23	4, 8, 22	syl2anc 596	. . . . . . . 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 4159	. . . . . . . . . . . 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 3186	. . . . . . . . . 10 ⊢ (𝑜 = ((int‘𝐽)‘𝑛) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅))
28	24, 27	imbi12d 347	. . . . . . . . 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 3573	. . . . . . . 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 18	. . . . . . 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 45	. . . . . 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 4187	. . . . . . . 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 418	. . . . . . . 8 ⊢ ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ⊆ (𝑛 ∩ (𝐹 “ 𝑠)) → ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
35	32, 34	syl 18	. . . . . . 7 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → ((((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
36	35	ralimdv 3177	. . . . . 6 ⊢ (((int‘𝐽)‘𝑛) ⊆ 𝑛 → (∀𝑠 ∈ 𝐿 (((int‘𝐽)‘𝑛) ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅))
37	10, 31, 36	sylsyld 62	. . . . 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 3163	. . . 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 1210	. . . . . . . . . 10 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
40	39, 3	syl 18	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
41		opnneip 23430	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
42	41	3expb 1138	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
43	40, 42	sylan 592	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ 𝑋) ∧ (𝑜 ∈ 𝐽 ∧ 𝐴 ∈ 𝑜)) → 𝑜 ∈ ((nei‘𝐽)‘{𝐴}))
44		ineq1 4159	. . . . . . . . . . 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 3186	. . . . . . . . 9 ⊢ (𝑛 = 𝑜 → (∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
47	46	rspcv 3573	. . . . . . . 8 ⊢ (𝑜 ∈ ((nei‘𝐽)‘{𝐴}) → (∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐿 (𝑛 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))
48	43, 47	syl 18	. . . . . . 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 462	. . . . . 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 87	. . . . 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 3163	. . . 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 215	. . 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 590	. 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 282	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)) ≠ ∅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ cuni 4867   “ cima 5654  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Topctop 23204  TopOnctopon 23221  intcnt 23328  neicnei 23408  Filcfil 24157   fClusf cfcf 24249

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-map 8842  df-fbas 21668  df-fg 21669  df-top 23205  df-topon 23222  df-cld 23330  df-ntr 23331  df-cls 23332  df-nei 23409  df-fil 24158  df-fm 24250  df-fcls 24253  df-fcf 24254

This theorem is used by:  fcfneii  24349