Theorem ustfilxp 24222

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 6943	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 24213	. . . . . . 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 267	. . . . 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 481	. . . 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 1142	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
7	5	simp2d 1143	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑋 × 𝑋) ∈ 𝑈)
8	7	ne0d 4341	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ≠ ∅)
9	5	simp3d 1144	. . . . . . . . . 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 3250	. . . . . . . . 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 1144	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))
12	11	simp1d 1142	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑣)
13		opelidres 6008	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋))
14	13	elv 3484	. . . . . . . . . . . 12 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋)
15	14	biimpri 228	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑋 → ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
16	15	rgen 3062	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)
17		r19.2z 4494	. . . . . . . . . 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 4340	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
21	20	rexlimivw 3150	. . . . . . . 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 4403	. . . . . . 7 ⊢ ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑣 ≠ ∅)
24	12, 22, 23	syl2anc 584	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → 𝑣 ≠ ∅)
25	24	nelrdva 3710	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ¬ ∅ ∈ 𝑈)
26		df-nel 3046	. . . . 5 ⊢ (∅ ∉ 𝑈 ↔ ¬ ∅ ∈ 𝑈)
27	25, 26	sylibr 234	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∅ ∉ 𝑈)
28	10	simp2d 1143	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈)
29	28	r19.21bi 3250	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝑈)
30		vex 3483	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
31	30	inex2 5317	. . . . . . . . . 10 ⊢ (𝑣 ∩ 𝑤) ∈ V
32	31	pwid 4621	. . . . . . . . 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 4199	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)))
35	34	ne0d 4341	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
36	35	ralrimiva 3145	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
37	36	ralrimiva 3145	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
38	8, 27, 37	3jca 1128	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))
39	1, 1	xpexd 7772	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ V)
40		isfbas 23838	. . . . 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 481	. . 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 4352	. . . . 5 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤))
45		elin 3966	. . . . . . 7 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤))
46		velpw 4604	. . . . . . . 8 ⊢ (𝑣 ∈ 𝒫 𝑤 ↔ 𝑣 ⊆ 𝑤)
47	46	anbi2i 623	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
48	45, 47	bitri 275	. . . . . 6 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
49	48	exbii 1847	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
50	44, 49	bitri 275	. . . 4 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
51	10	simp1d 1142	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
52	51	r19.21bi 3250	. . . . . . 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 453	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
55	54	exlimdv 1932	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
56	50, 55	biimtrid 242	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
57	56	ralrimiva 3145	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
58		isfil 23856	. 2 ⊢ (𝑈 ∈ (Fil‘(𝑋 × 𝑋)) ↔ (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈)))
59	43, 57, 58	sylanbrc 583	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∃wex 1778   ∈ wcel 2107   ≠ wne 2939   ∉ wnel 3045  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  𝒫 cpw 4599  ⟨cop 4631   I cid 5576   × cxp 5682  ^◡ccnv 5683   ↾ cres 5686   ∘ ccom 5688  ‘cfv 6560  fBascfbas 21353  Filcfil 23854  UnifOncust 24209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-fbas 21362  df-fil 23855  df-ust 24210

This theorem is referenced by: (None)