Theorem ustfilxp 23110

Description: A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
ustfilxp	⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Proof of Theorem ustfilxp

Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6750	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 23101	. . . . . . 7 ⊢ (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))))
4	3	ibi 270	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
5	4	adantl 485	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))))
6	5	simp1d 1144	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
7	5	simp2d 1145	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑋 × 𝑋) ∈ 𝑈)
8	7	ne0d 4250	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ≠ ∅)
9	5	simp3d 1146	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))
10	9	r19.21bi 3130	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)))
11	10	simp3d 1146	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))
12	11	simp1d 1144	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑣)
13		opelidres 5863	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋))
14	13	elv 3414	. . . . . . . . . . . 12 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋)
15	14	biimpri 231	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑋 → ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
16	15	rgen 3071	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)
17		r19.2z 4406	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
18	16, 17	mpan2 691	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
19	18	ad2antrr 726	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
20		ne0i 4249	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
21	20	rexlimivw 3201	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
22	19, 21	syl 17	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ≠ ∅)
23		ssn0 4315	. . . . . . 7 ⊢ ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑣 ≠ ∅)
24	12, 22, 23	syl2anc 587	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → 𝑣 ≠ ∅)
25	24	nelrdva 3618	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ¬ ∅ ∈ 𝑈)
26		df-nel 3047	. . . . 5 ⊢ (∅ ∉ 𝑈 ↔ ¬ ∅ ∈ 𝑈)
27	25, 26	sylibr 237	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∅ ∉ 𝑈)
28	10	simp2d 1145	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈)
29	28	r19.21bi 3130	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝑈)
30		vex 3412	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
31	30	inex2 5211	. . . . . . . . . 10 ⊢ (𝑣 ∩ 𝑤) ∈ V
32	31	pwid 4537	. . . . . . . . 9 ⊢ (𝑣 ∩ 𝑤) ∈ 𝒫 (𝑣 ∩ 𝑤)
33	32	a1i 11	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝒫 (𝑣 ∩ 𝑤))
34	29, 33	elind 4108	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)))
35	34	ne0d 4250	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
36	35	ralrimiva 3105	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
37	36	ralrimiva 3105	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
38	8, 27, 37	3jca 1130	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))
39	1, 1	xpexd 7536	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ V)
40		isfbas 22726	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ V → (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))))
41	39, 40	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))))
42	41	adantl 485	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))))
43	6, 38, 42	mpbir2and 713	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (fBas‘(𝑋 × 𝑋)))
44		n0 4261	. . . . 5 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤))
45		elin 3882	. . . . . . 7 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤))
46		velpw 4518	. . . . . . . 8 ⊢ (𝑣 ∈ 𝒫 𝑤 ↔ 𝑣 ⊆ 𝑤)
47	46	anbi2i 626	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
48	45, 47	bitri 278	. . . . . 6 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
49	48	exbii 1855	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
50	44, 49	bitri 278	. . . 4 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
51	10	simp1d 1144	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
52	51	r19.21bi 3130	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
53	52	an32s 652	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ∈ 𝑈) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
54	53	expimpd 457	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
55	54	exlimdv 1941	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
56	50, 55	syl5bi 245	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
57	56	ralrimiva 3105	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
58		isfil 22744	. 2 ⊢ (𝑈 ∈ (Fil‘(𝑋 × 𝑋)) ↔ (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈)))
59	43, 57, 58	sylanbrc 586	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∃wex 1787   ∈ wcel 2110   ≠ wne 2940   ∉ wnel 3046  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237  𝒫 cpw 4513  ⟨cop 4547   I cid 5454   × cxp 5549  ^◡ccnv 5550   ↾ cres 5553   ∘ ccom 5555  ‘cfv 6380  fBascfbas 20351  Filcfil 22742  UnifOncust 23097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-fbas 20360  df-fil 22743  df-ust 23098

This theorem is referenced by: (None)