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Theorem hausflim 24312

Description: A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

**Hypothesis**
Ref	Expression
flimcf.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
hausflim	⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))

Distinct variable groups: 𝑥,𝑓,𝐽 𝑓,𝑋,𝑥

Proof of Theorem hausflim Dummy variables 𝑣 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		haustop 23661	. . 3 ⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top)
2		hausflimi 24311	. . . 4 ⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
3	2	ralrimivw 3161	. . 3 ⊢ (𝐽 ∈ Haus → ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
4	1, 3	jca 521	. 2 ⊢ (𝐽 ∈ Haus → (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
5		flimcf.1	. . . . . . . . . . . . . . 15 ⊢ 𝑋 = ∪ 𝐽
6	5	toptopon 23247	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
7	6	birani 509	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝐽 ∈ (TopOn‘𝑋))
8		simprll 791	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑧 ∈ 𝑋)
9	8	snssd 4746	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ⊆ 𝑋)
10	8	snn0d 4735	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ≠ ∅)
11		neifil 24211	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑧} ⊆ 𝑋 ∧ {𝑧} ≠ ∅) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
12	7, 9, 10, 11	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋))
13		filfbas 24179	. . . . . . . . . . . 12 ⊢ (((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋))
14	12, 13	syl 18	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋))
15		simprlr 792	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑤 ∈ 𝑋)
16	15	snssd 4746	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ⊆ 𝑋)
17	15	snn0d 4735	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ≠ ∅)
18		neifil 24211	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑤} ⊆ 𝑋 ∧ {𝑤} ≠ ∅) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
19	7, 16, 17, 18	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋))
20		filfbas 24179	. . . . . . . . . . . 12 ⊢ (((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋))
21	19, 20	syl 18	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋))
22		fbunfip 24200	. . . . . . . . . . 11 ⊢ ((((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋) ∧ ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅))
23	14, 21, 22	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅))
24	5	neisspw 23437	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑧}) ⊆ 𝒫 𝑋)
25	5	neisspw 23437	. . . . . . . . . . . . . . 15 ⊢ (𝐽 ∈ Top → ((nei‘𝐽)‘{𝑤}) ⊆ 𝒫 𝑋)
26	24, 25	unssd 4192	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋)
27	26	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋)
28	27	a1d 26	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋))
29		ssun1 4178	. . . . . . . . . . . . . 14 ⊢ ((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))
30		filn0 24193	. . . . . . . . . . . . . . 15 ⊢ (((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ≠ ∅)
31	12, 30	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ≠ ∅)
32		ssn0 4357	. . . . . . . . . . . . . 14 ⊢ ((((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∧ ((nei‘𝐽)‘{𝑧}) ≠ ∅) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)
33	29, 31, 32	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)
34	33	a1d 26	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅))
35		idd 25	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
36	28, 34, 35	3jcad 1147	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
37	5	topopn 23236	. . . . . . . . . . . . . 14 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
38	37	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑋 ∈ 𝐽)
39		fsubbas 24198	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
40	38, 39	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
41		fgcl 24209	. . . . . . . . . . . . . . 15 ⊢ ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋))
42	41	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋))
43		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ≠ 𝑤)
44	8	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ 𝑋)
45	15	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ 𝑋)
46		fvex 6906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑧}) ∈ V
47		fvex 6906	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((nei‘𝐽)‘{𝑤}) ∈ V
48	46, 47	unex 7768	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V
49		ssfii 9421	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))
50	48, 49	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))
51		ssfg 24203	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
52	51	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
53	50, 52	sstrid 3923	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
54	29, 53	sstrid 3923	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
55	7	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
56		elflim 24302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
57	55, 42, 56	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
58	44, 54, 57	mpbir2and 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
59	53	unssbd 4194	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))
60		elflim 24302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
61	55, 42, 60	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
62	45, 59, 61	mpbir2and 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
63		eleq1w 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
64		eleq1w 2846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑤 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
65	63, 64	moi 3678	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → 𝑧 = 𝑤)
66	65	3com23 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → 𝑧 = 𝑤)
67	66	3expia 1139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤))
68	44, 45, 58, 62, 67	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤))
69	68	necon3ad 2971	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ≠ 𝑤 → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
70	43, 69	mpd 16	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
71		oveq2 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝐽 fLim 𝑓) = (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))
72	71	eleq2d 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
73	72	mobidv 2577	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
74	73	notbid 321	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))))
75	74	rspcev 3579	. . . . . . . . . . . . . 14 ⊢ (((𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋) ∧ ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
76	42, 70, 75	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
77	76	ex 418	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
78	40, 77	sylbird 263	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
79	36, 78	syld 48	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
80	23, 79	sylbird 263	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
81		df-ne 2959	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ (𝑢 ∩ 𝑣) = ∅)
82	81	ralbii 3111	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅)
83		ralnex 3091	. . . . . . . . . . . 12 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
84	82, 83	bitri 278	. . . . . . . . . . 11 ⊢ (∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
85	84	ralbii 3111	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
86		ralnex 3091	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
87	85, 86	bitri 278	. . . . . . . . 9 ⊢ (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
88		rexnal 3117	. . . . . . . . 9 ⊢ (∃𝑓 ∈ (Fil‘𝑋) ¬ ∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∀𝑓 ∈ (Fil‘𝑋)∃𝑥 𝑥 ∈ (𝐽 fLim 𝑓))
89	80, 87, 88	3imtr3g 298	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ → ¬ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))
90	89	con4d 116	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
91	90	imp 412	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
92	91	an32s 665	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)
93	92	expr 462	. . . 4 ⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
94	93	ralrimivva 3208	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))
95	6	birani 509	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
96		hausnei2 23683	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)))
97	95, 96	syl 18	. . 3 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)))
98	94, 97	mpbird 260	. 2 ⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ Haus)
99	4, 98	impbii 212	1 ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ∃*wmo 2565 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 Vcvv 3453 ⊆ wss 3899 ∩ cin 4091 ∪ cun 4151 ∅c0 4280 𝒫 cpw 4559 {csn 4587 ∪ cuni 4869 ‘cfv 6546 (class class class)co 7427 ficfi 9412 fBascfbas 21678 filGencfg 21679 Topctop 23223 TopOnctopon 23240 neicnei 23427 Hauscha 23638 Filcfil 24176 fLim cflim 24265

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5234 ax-sep 5251 ax-nul 5262 ax-pow 5330 ax-pr 5394 ax-un 7758

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-reu 3369 df-rab 3416 df-v 3455 df-sbc 3742 df-csb 3851 df-ss 3900 df-pss 4016 df-dif 4049 df-in 4093 df-un 4153 df-nul 4281 df-if 4485 df-pw 4561 df-sn 4588 df-pr 4590 df-op 4838 df-uni 4870 df-int 4910 df-iun 4955 df-br 5106 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5549 df-eprel 5554 df-po 5562 df-so 5563 df-fr 5607 df-we 5609 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6500 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7430 df-oprab 7431 df-mpo 7432 df-om 7885 df-1o 8487 df-2o 8488 df-en 8985 df-fin 8988 df-fi 9413 df-fbas 21687 df-fg 21688 df-top 23224 df-topon 23241 df-nei 23428 df-haus 23645 df-fil 24177 df-flim 24270

This theorem is used by: (None)

W3C validator