Theorem ustfilxp 22821

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 6703	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 22812	. . . . . . 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 269	. . . . 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 484	. . . 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 1138	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
7	5	simp2d 1139	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑋 × 𝑋) ∈ 𝑈)
8	7	ne0d 4301	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ≠ ∅)
9	5	simp3d 1140	. . . . . . . . . 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 3208	. . . . . . . . 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 1140	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))
12	11	simp1d 1138	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑣)
13		opelidres 5865	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋))
14	13	elv 3499	. . . . . . . . . . . 12 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋)
15	14	biimpri 230	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑋 → ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
16	15	rgen 3148	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)
17		r19.2z 4440	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
18	16, 17	mpan2 689	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
19	18	ad2antrr 724	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
20		ne0i 4300	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
21	20	rexlimivw 3282	. . . . . . . 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 4354	. . . . . . 7 ⊢ ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑣 ≠ ∅)
24	12, 22, 23	syl2anc 586	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → 𝑣 ≠ ∅)
25	24	nelrdva 3696	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ¬ ∅ ∈ 𝑈)
26		df-nel 3124	. . . . 5 ⊢ (∅ ∉ 𝑈 ↔ ¬ ∅ ∈ 𝑈)
27	25, 26	sylibr 236	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∅ ∉ 𝑈)
28	10	simp2d 1139	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈)
29	28	r19.21bi 3208	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝑈)
30		vex 3497	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
31	30	inex2 5222	. . . . . . . . . 10 ⊢ (𝑣 ∩ 𝑤) ∈ V
32	31	pwid 4563	. . . . . . . . 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 4171	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)))
35	34	ne0d 4301	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
36	35	ralrimiva 3182	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
37	36	ralrimiva 3182	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
38	8, 27, 37	3jca 1124	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))
39	1, 1	xpexd 7474	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ V)
40		isfbas 22437	. . . . 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 484	. . 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 711	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (fBas‘(𝑋 × 𝑋)))
44		n0 4310	. . . . 5 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤))
45		elin 4169	. . . . . . 7 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤))
46		velpw 4544	. . . . . . . 8 ⊢ (𝑣 ∈ 𝒫 𝑤 ↔ 𝑣 ⊆ 𝑤)
47	46	anbi2i 624	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
48	45, 47	bitri 277	. . . . . 6 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
49	48	exbii 1848	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
50	44, 49	bitri 277	. . . 4 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
51	10	simp1d 1138	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
52	51	r19.21bi 3208	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
53	52	an32s 650	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ∈ 𝑈) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
54	53	expimpd 456	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
55	54	exlimdv 1934	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
56	50, 55	syl5bi 244	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
57	56	ralrimiva 3182	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
58		isfil 22455	. 2 ⊢ (𝑈 ∈ (Fil‘(𝑋 × 𝑋)) ↔ (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈)))
59	43, 57, 58	sylanbrc 585	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016   ∉ wnel 3123  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  ⟨cop 4573   I cid 5459   × cxp 5553  ^◡ccnv 5554   ↾ cres 5557   ∘ ccom 5559  ‘cfv 6355  fBascfbas 20533  Filcfil 22453  UnifOncust 22808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-fbas 20542  df-fil 22454  df-ust 22809

This theorem is referenced by: (None)