Theorem utoptop 22846

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 487	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ (unifTop‘𝑈))
2		utopval 22844	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎})
3		ssrab2 4059	. . . . . . . . 9 ⊢ {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑝 ∈ 𝑎 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑎} ⊆ 𝒫 𝑋
4	2, 3	eqsstrdi 4024	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
5	4	adantr 483	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (unifTop‘𝑈) ⊆ 𝒫 𝑋)
6	1, 5	sstrd 3980	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → 𝑥 ⊆ 𝒫 𝑋)
7		sspwuni 5025	. . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑋 ↔ ∪ 𝑥 ⊆ 𝑋)
8	6, 7	sylib 220	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∪ 𝑥 ⊆ 𝑋)
9		simp-4l 781	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑈 ∈ (UnifOn‘𝑋))
10		simp-4r 782	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑥 ⊆ (unifTop‘𝑈))
11		simplr 767	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑦 ∈ 𝑥)
12	10, 11	sseldd 3971	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑦 ∈ (unifTop‘𝑈))
13		simpr 487	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → 𝑝 ∈ 𝑦)
14		elutop 22845	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑦 ∈ (unifTop‘𝑈) ↔ (𝑦 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)))
15	14	biimpa 479	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → (𝑦 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
16	15	simprd 498	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) → ∀𝑝 ∈ 𝑦 ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
17	16	r19.21bi 3211	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑦 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
18	9, 12, 13, 17	syl21anc 835	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
19		r19.41v 3350	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) ↔ (∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥))
20		ssuni 4866	. . . . . . . . . 10 ⊢ (((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
21	20	reximi 3246	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑈 ((𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
22	19, 21	sylbir 237	. . . . . . . 8 ⊢ ((∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
23	18, 11, 22	syl2anc 586	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) ∧ 𝑦 ∈ 𝑥) ∧ 𝑝 ∈ 𝑦) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
24		eluni2 4845	. . . . . . . . 9 ⊢ (𝑝 ∈ ∪ 𝑥 ↔ ∃𝑦 ∈ 𝑥 𝑝 ∈ 𝑦)
25	24	biimpi 218	. . . . . . . 8 ⊢ (𝑝 ∈ ∪ 𝑥 → ∃𝑦 ∈ 𝑥 𝑝 ∈ 𝑦)
26	25	adantl 484	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) → ∃𝑦 ∈ 𝑥 𝑝 ∈ 𝑦)
27	23, 26	r19.29a 3292	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) ∧ 𝑝 ∈ ∪ 𝑥) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
28	27	ralrimiva 3185	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)
29		elutop 22845	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝑥 ∈ (unifTop‘𝑈) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)))
30	29	adantr 483	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → (∪ 𝑥 ∈ (unifTop‘𝑈) ↔ (∪ 𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ ∪ 𝑥∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ ∪ 𝑥)))
31	8, 28, 30	mpbir2and 711	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ⊆ (unifTop‘𝑈)) → ∪ 𝑥 ∈ (unifTop‘𝑈))
32	31	ex 415	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)))
33	32	alrimiv 1927	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)))
34		elutop 22845	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑥 ∈ (unifTop‘𝑈) ↔ (𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)))
35	34	biimpa 479	. . . . . . 7 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → (𝑥 ⊆ 𝑋 ∧ ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥))
36	35	simpld 497	. . . . . 6 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → 𝑥 ⊆ 𝑋)
37	36	adantrr 715	. . . . 5 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → 𝑥 ⊆ 𝑋)
38		ssinss1 4217	. . . . 5 ⊢ (𝑥 ⊆ 𝑋 → (𝑥 ∩ 𝑦) ⊆ 𝑋)
39	37, 38	syl 17	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥 ∩ 𝑦) ⊆ 𝑋)
40		simpl 485	. . . . . . . . . 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))
41		simpr31 1259	. . . . . . . . . 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)
42		simpr32 1260	. . . . . . . . . 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)
43		ustincl 22819	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈) → (𝑢 ∩ 𝑣) ∈ 𝑈)
44	40, 41, 42, 43	syl3anc 1367	. . . . . . . . 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)
45		inss1 4208	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑢
46		imass1 5967	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑢 → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝}))
47	45, 46	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑢 “ {𝑝})
48		simpr33 1261	. . . . . . . . . . . 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))
49	48	simpld 497	. . . . . . . . . . 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)
50	47, 49	sstrid 3981	. . . . . . . . . 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)
51		inss2 4209	. . . . . . . . . . . 12 ⊢ (𝑢 ∩ 𝑣) ⊆ 𝑣
52		imass1 5967	. . . . . . . . . . . 12 ⊢ ((𝑢 ∩ 𝑣) ⊆ 𝑣 → ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝}))
53	51, 52	ax-mp 5	. . . . . . . . . . 11 ⊢ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑣 “ {𝑝})
54	48	simprd 498	. . . . . . . . . . 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)
55	53, 54	sstrid 3981	. . . . . . . . . 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)
56	50, 55	ssind 4212	. . . . . . . . 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))
57		imaeq1 5927	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑢 ∩ 𝑣) → (𝑤 “ {𝑝}) = ((𝑢 ∩ 𝑣) “ {𝑝}))
58	57	sseq1d 4001	. . . . . . . . . 10 ⊢ (𝑤 = (𝑢 ∩ 𝑣) → ((𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦) ↔ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑥 ∩ 𝑦)))
59	58	rspcev 3626	. . . . . . . . 9 ⊢ (((𝑢 ∩ 𝑣) ∈ 𝑈 ∧ ((𝑢 ∩ 𝑣) “ {𝑝}) ⊆ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
60	44, 56, 59	syl2anc 586	. . . . . . . 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))
61	60	3anassrs 1356	. . . . . . 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))
62	61	3anassrs 1356	. . . . . 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))
63		simpll 765	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑈 ∈ (UnifOn‘𝑋))
64		simplrl 775	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑥 ∈ (unifTop‘𝑈))
65		simpr 487	. . . . . . . . . 10 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑝 ∈ (𝑥 ∩ 𝑦))
66		elin 4172	. . . . . . . . . 10 ⊢ (𝑝 ∈ (𝑥 ∩ 𝑦) ↔ (𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦))
67	65, 66	sylib 220	. . . . . . . . 9 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → (𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦))
68	67	simpld 497	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑝 ∈ 𝑥)
69	35	simprd 498	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) → ∀𝑝 ∈ 𝑥 ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
70	69	r19.21bi 3211	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑥 ∈ (unifTop‘𝑈)) ∧ 𝑝 ∈ 𝑥) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
71	63, 64, 68, 70	syl21anc 835	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥)
72		simplrr 776	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑦 ∈ (unifTop‘𝑈))
73	67	simprd 498	. . . . . . . 8 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → 𝑝 ∈ 𝑦)
74	63, 72, 73, 17	syl21anc 835	. . . . . . 7 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦)
75		reeanv 3370	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦) ↔ (∃𝑢 ∈ 𝑈 (𝑢 “ {𝑝}) ⊆ 𝑥 ∧ ∃𝑣 ∈ 𝑈 (𝑣 “ {𝑝}) ⊆ 𝑦))
76	71, 74, 75	sylanbrc 585	. . . . . 6 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑢 ∈ 𝑈 ∃𝑣 ∈ 𝑈 ((𝑢 “ {𝑝}) ⊆ 𝑥 ∧ (𝑣 “ {𝑝}) ⊆ 𝑦))
77	62, 76	r19.29vva 3339	. . . . 5 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) ∧ 𝑝 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
78	77	ralrimiva 3185	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → ∀𝑝 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))
79		elutop 22845	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈) ↔ ((𝑥 ∩ 𝑦) ⊆ 𝑋 ∧ ∀𝑝 ∈ (𝑥 ∩ 𝑦)∃𝑤 ∈ 𝑈 (𝑤 “ {𝑝}) ⊆ (𝑥 ∩ 𝑦))))
80	79	adantr 483	. . . 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))))
81	39, 78, 80	mpbir2and 711	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑥 ∈ (unifTop‘𝑈) ∧ 𝑦 ∈ (unifTop‘𝑈))) → (𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))
82	81	ralrimivva 3194	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))
83		fvex 6686	. . 3 ⊢ (unifTop‘𝑈) ∈ V
84		istopg 21506	. . 3 ⊢ ((unifTop‘𝑈) ∈ V → ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈))))
85	83, 84	ax-mp 5	. 2 ⊢ ((unifTop‘𝑈) ∈ Top ↔ (∀𝑥(𝑥 ⊆ (unifTop‘𝑈) → ∪ 𝑥 ∈ (unifTop‘𝑈)) ∧ ∀𝑥 ∈ (unifTop‘𝑈)∀𝑦 ∈ (unifTop‘𝑈)(𝑥 ∩ 𝑦) ∈ (unifTop‘𝑈)))
86	33, 82, 85	sylanbrc 585	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1534   = wceq 1536   ∈ wcel 2113  ∀wral 3141  ∃wrex 3142  {crab 3145  Vcvv 3497   ∩ cin 3938   ⊆ wss 3939  𝒫 cpw 4542  {csn 4570  ∪ cuni 4841   “ cima 5561  ‘cfv 6358  Topctop 21504  UnifOncust 22811  unifTopcutop 22842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366  df-top 21505  df-ust 22812  df-utop 22843

This theorem is referenced by:  utoptopon  22848  utop2nei  22862  utop3cls  22863  utopreg  22864