Theorem utoptop 24553

Description: The topology induced by a uniform structure 𝑈 is a topology. (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
utoptop	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)

Proof of Theorem utoptop

Dummy variables 𝑝 𝑎 𝑢 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 490	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2		utopval 24551	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3		ssrab2 4005	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
4	2, 3	eqsstrdi 3952	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
5	4	adantr 486	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
6	1, 5	sstrd 3918	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7		sspwuni 5059	. . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑋 ↔ ∪ 𝑥 ⊆ 𝑋)
8	6, 7	sylib 221	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∪ 𝑥 ⊆ 𝑋)
9		simp-4l 795	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10		simp-4r 796	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11		simplr 781	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑦 ∈ 𝑥)
12	10, 11	sseldd 3909	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13		simpr 490	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑝 ∈ 𝑦)
14		elutop 24552	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
15	14	biimpa 482	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
16	15	simprd 501	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
17	16	r19.21bi 3254	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
18	9, 12, 13, 17	syl21anc 851	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
19		r19.41v 3192	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) ↔ (∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥))
20		ssuni 4892	. . . . . . . . . 10 ⊢ (((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
21	20	reximi 3100	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
22	19, 21	sylbir 238	. . . . . . . 8 ⊢ ((∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
23	18, 11, 22	syl2anc 596	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
24		eluni2 4870	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑝 ∈ 𝑦)
25	24	bilani 510	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) → ∃𝑦 ∈ 𝑥 𝑝 ∈ 𝑦)
26	23, 25	r19.29a 3170	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
27	26	ralrimiva 3154	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
28		elutop 24552	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝑥 ∈ (unifTop‘𝑈) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)))
29	28	adantr 486	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (∪ 𝑥 ∈ (unifTop‘𝑈) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)))
30	8, 27, 29	mpbir2and 726	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∪ 𝑥 ∈ (unifTop‘𝑈))
31	30	ex 418	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)))
32	31	alrimiv 1959	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)))
33		elutop 24552	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
34	33	biimpa 482	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
35	34	simpld 500	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥 ⊆ 𝑋)
36	35	adantrr 730	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥 ⊆ 𝑋)
37		ssinss1 4142	. . . . 5 ⊢ (𝑥 ⊆ 𝑋 → (𝑥 ∩ 𝑦) ⊆ 𝑋)
38	36, 37	syl 18	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥 ∩ 𝑦) ⊆ 𝑋)
39		simpl 488	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑈 ∈ (UnifOn‘𝑋))
40		simpr31 1282	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑢 ∈ 𝑈)
41		simpr32 1283	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → 𝑣 ∈ 𝑈)
42		ustincl 24527	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈) → (𝑢 ∩ 𝑣) ∈ 𝑈)
43	39, 40, 41, 42	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 ∩ 𝑣) ∈ 𝑈)
44		inss1 4133	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
45		imass1 6053	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑢 → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝}))
46	44, 45	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝})
47		simpr33 1284	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
48	47	simpld 500	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑢 “ {𝑝}) ⊆ 𝑥)
49	46, 48	sstrid 3919	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ 𝑥)
50		inss2 4134	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑣
51		imass1 6053	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑣 → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
53	47	simprd 501	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → (𝑣 “ {𝑝}) ⊆ 𝑦)
54	52, 53	sstrid 3919	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ 𝑦)
55	49, 54	ssind 4137	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
56		imaeq1 6046	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑢 ∩ 𝑣) → (𝑤 “ {𝑝}) = ((𝑢 ∩ 𝑣) “ {𝑝}))
57	56	sseq1d 3939	. . . . . . . . . 10 ⊢ (𝑤 = (𝑢 ∩ 𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦) ↔ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑥 ∩ 𝑦)))
58	57	rspcev 3576	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝑈 ∧ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
59	43, 55, 58	syl2anc 596	. . . . . . . 8 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ ((𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)))) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
60	59	3anassrs 1381	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
61	60	3anassrs 1381	. . . . . 6 ⊢ ((((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) ∧ 𝑢 ∈ 𝑈) ∧ 𝑣 ∈ 𝑈) ∧ ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦)) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
62		simpll 779	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
63		simplrl 789	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
64		elin 4093	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝑥 ∩ 𝑦) ↔ (𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦))
65	64	bilani 510	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → (𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦))
66	65	simpld 500	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑝 ∈ 𝑥)
67	34	simprd 501	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
68	67	r19.21bi 3254	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ 𝑥) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
69	62, 63, 66, 68	syl21anc 851	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
70		simplrr 790	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
71	65	simprd 501	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑝 ∈ 𝑦)
72	62, 70, 71, 17	syl21anc 851	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
73		reeanv 3234	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
74	69, 72, 73	sylanbrc 595	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
75	61, 74	r19.29vva 3222	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
76	75	ralrimiva 3154	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
77		elutop 24552	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥 ∩ 𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))))
78	77	adantr 486	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ((𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥 ∩ 𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))))
79	38, 76, 78	mpbir2and 726	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))
80	79	ralrimivva 3205	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))
81		fvex 6898	. . 3 ⊢ (unifTop‘𝑈) ∈ V
82		istopg 23213	. . 3 ⊢ ((unifTop‘𝑈) ∈ V → ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))))
83	81, 82	ax-mp 5	. 2 ⊢ ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈)))
84	32, 80, 83	sylanbrc 595	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∩ cin 4089  𝒫 cpw 4556  {csn 4583  ∪ cuni 4866   “ cima 5653  ‘cfv 6538  Topctop 23211  UnifOncust 24519  unifTopcutop 24549

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 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-top 23212  df-ust 24520  df-utop 24550

This theorem is used by:  utoptopon  24555  utop2nei  24569  utop3cls  24570  utopreg  24571