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Theorem ustfilxp 23937

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 6928	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 23928	. . . . . . 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 266	. . . . 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 480	. . . 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 1140	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
7	5	simp2d 1141	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑋 × 𝑋) ∈ 𝑈)
8	7	ne0d 4334	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ≠ ∅)
9	5	simp3d 1142	. . . . . . . . . 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 3246	. . . . . . . . 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 1142	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))
12	11	simp1d 1140	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑣)
13		opelidres 5992	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋))
14	13	elv 3478	. . . . . . . . . . . 12 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋)
15	14	biimpri 227	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑋 → ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
16	15	rgen 3061	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)
17		r19.2z 4493	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
18	16, 17	mpan2 687	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
19	18	ad2antrr 722	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
20		ne0i 4333	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
21	20	rexlimivw 3149	. . . . . . . 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 4399	. . . . . . 7 ⊢ ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑣 ≠ ∅)
24	12, 22, 23	syl2anc 582	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → 𝑣 ≠ ∅)
25	24	nelrdva 3700	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ¬ ∅ ∈ 𝑈)
26		df-nel 3045	. . . . 5 ⊢ (∅ ∉ 𝑈 ↔ ¬ ∅ ∈ 𝑈)
27	25, 26	sylibr 233	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∅ ∉ 𝑈)
28	10	simp2d 1141	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈)
29	28	r19.21bi 3246	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝑈)
30		vex 3476	. . . . . . . . . . 11 ⊢ 𝑤 ∈ V
31	30	inex2 5317	. . . . . . . . . 10 ⊢ (𝑣 ∩ 𝑤) ∈ V
32	31	pwid 4623	. . . . . . . . 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 4193	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)))
35	34	ne0d 4334	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
36	35	ralrimiva 3144	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
37	36	ralrimiva 3144	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
38	8, 27, 37	3jca 1126	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))
39	1, 1	xpexd 7740	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ V)
40		isfbas 23553	. . . . 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 480	. . 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 709	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (fBas‘(𝑋 × 𝑋)))
44		n0 4345	. . . . 5 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤))
45		elin 3963	. . . . . . 7 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤))
46		velpw 4606	. . . . . . . 8 ⊢ (𝑣 ∈ 𝒫 𝑤 ↔ 𝑣 ⊆ 𝑤)
47	46	anbi2i 621	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
48	45, 47	bitri 274	. . . . . 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 274	. . . 4 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
51	10	simp1d 1140	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
52	51	r19.21bi 3246	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
53	52	an32s 648	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ∈ 𝑈) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
54	53	expimpd 452	. . . . 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	biimtrid 241	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
57	56	ralrimiva 3144	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
58		isfil 23571	. 2 ⊢ (𝑈 ∈ (Fil‘(𝑋 × 𝑋)) ↔ (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈)))
59	43, 57, 58	sylanbrc 581	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 ∃wex 1779 ∈ wcel 2104 ≠ wne 2938 ∉ wnel 3044 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ⟨cop 4633 I cid 5572 × cxp 5673 ^◡ccnv 5674 ↾ cres 5677 ∘ ccom 5679 ‘cfv 6542 fBascfbas 21132 Filcfil 23569 UnifOncust 23924

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fv 6550 df-fbas 21141 df-fil 23570 df-ust 23925

This theorem is referenced by: (None)

W3C validator