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Theorem ustfilxp 21826

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.)

**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 6131	. . . . . . 7 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 21817	. . . . . . 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 255	. . . . 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 1066	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
7	5	simp2d 1067	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑋 × 𝑋) ∈ 𝑈)
8		ne0i 3880	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → 𝑈 ≠ ∅)
9	7, 8	syl 17	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ≠ ∅)
10	5	simp3d 1068	. . . . . . . . . 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)))
11	10	r19.21bi 2916	. . . . . . . . 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)))
12	11	simp3d 1068	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → (( I ↾ 𝑋) ⊆ 𝑣 ∧ ^◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣))
13	12	simp1d 1066	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ⊆ 𝑣)
14		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
15		opelresi 5328	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ V → (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) ↔ 𝑤 ∈ 𝑋)
17	16	biimpri 217	. . . . . . . . . . 11 ⊢ (𝑤 ∈ 𝑋 → ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
18	17	rgen 2906	. . . . . . . . . 10 ⊢ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)
19		r19.2z 4012	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ ∀𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋)) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
20	18, 19	mpan2 703	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
21	20	ad2antrr 758	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋))
22		ne0i 3880	. . . . . . . . 9 ⊢ (⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
23	22	rexlimivw 3011	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝑋 ⟨𝑤, 𝑤⟩ ∈ ( I ↾ 𝑋) → ( I ↾ 𝑋) ≠ ∅)
24	21, 23	syl 17	. . . . . . 7 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ( I ↾ 𝑋) ≠ ∅)
25		ssn0 3928	. . . . . . 7 ⊢ ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ( I ↾ 𝑋) ≠ ∅) → 𝑣 ≠ ∅)
26	13, 24, 25	syl2anc 691	. . . . . 6 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → 𝑣 ≠ ∅)
27	26	nelrdva 3384	. . . . 5 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ¬ ∅ ∈ 𝑈)
28		df-nel 2783	. . . . 5 ⊢ (∅ ∉ 𝑈 ↔ ¬ ∅ ∈ 𝑈)
29	27, 28	sylibr 223	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∅ ∉ 𝑈)
30	11	simp2d 1067	. . . . . . . . 9 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈)
31	30	r19.21bi 2916	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝑈)
32	14	inex2 4728	. . . . . . . . . 10 ⊢ (𝑣 ∩ 𝑤) ∈ V
33	32	pwid 4122	. . . . . . . . 9 ⊢ (𝑣 ∩ 𝑤) ∈ 𝒫 (𝑣 ∩ 𝑤)
34	33	a1i 11	. . . . . . . 8 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ 𝒫 (𝑣 ∩ 𝑤))
35	31, 34	elind 3760	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)))
36		ne0i 3880	. . . . . . 7 ⊢ ((𝑣 ∩ 𝑤) ∈ (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
37	35, 36	syl 17	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝑈) → (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
38	37	ralrimiva 2949	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
39	38	ralrimiva 2949	. . . 4 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅)
40	9, 29, 39	3jca 1235	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))
41		xpexg 6858	. . . . . 6 ⊢ ((𝑋 ∈ V ∧ 𝑋 ∈ V) → (𝑋 × 𝑋) ∈ V)
42	1, 1, 41	syl2anc 691	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ V)
43		isfbas 21443	. . . . 5 ⊢ ((𝑋 × 𝑋) ∈ V → (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))))
44	42, 43	syl 17	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑈 ≠ ∅ ∧ ∅ ∉ 𝑈 ∧ ∀𝑣 ∈ 𝑈 ∀𝑤 ∈ 𝑈 (𝑈 ∩ 𝒫 (𝑣 ∩ 𝑤)) ≠ ∅))))
45	44	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)) ≠ ∅))))
46	6, 40, 45	mpbir2and 959	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (fBas‘(𝑋 × 𝑋)))
47		n0 3890	. . . . 5 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤))
48		elin 3758	. . . . . . 7 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤))
49		selpw 4115	. . . . . . . 8 ⊢ (𝑣 ∈ 𝒫 𝑤 ↔ 𝑣 ⊆ 𝑤)
50	49	anbi2i 726	. . . . . . 7 ⊢ ((𝑣 ∈ 𝑈 ∧ 𝑣 ∈ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
51	48, 50	bitri 263	. . . . . 6 ⊢ (𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ (𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
52	51	exbii 1764	. . . . 5 ⊢ (∃𝑣 𝑣 ∈ (𝑈 ∩ 𝒫 𝑤) ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
53	47, 52	bitri 263	. . . 4 ⊢ ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ ↔ ∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤))
54	11	simp1d 1066	. . . . . . . 8 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
55	54	r19.21bi 2916	. . . . . . 7 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑣 ∈ 𝑈) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
56	55	an32s 842	. . . . . 6 ⊢ ((((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) ∧ 𝑣 ∈ 𝑈) → (𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈))
57	56	expimpd 627	. . . . 5 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
58	57	exlimdv 1848	. . . 4 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → (∃𝑣(𝑣 ∈ 𝑈 ∧ 𝑣 ⊆ 𝑤) → 𝑤 ∈ 𝑈))
59	53, 58	syl5bi 231	. . 3 ⊢ (((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) ∧ 𝑤 ∈ 𝒫 (𝑋 × 𝑋)) → ((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
60	59	ralrimiva 2949	. 2 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈))
61		isfil 21461	. 2 ⊢ (𝑈 ∈ (Fil‘(𝑋 × 𝑋)) ↔ (𝑈 ∈ (fBas‘(𝑋 × 𝑋)) ∧ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)((𝑈 ∩ 𝒫 𝑤) ≠ ∅ → 𝑤 ∈ 𝑈)))
62	46, 60, 61	sylanbrc 695	1 ⊢ ((𝑋 ≠ ∅ ∧ 𝑈 ∈ (UnifOn‘𝑋)) → 𝑈 ∈ (Fil‘(𝑋 × 𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∀wral 2896 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ⟨cop 4131 I cid 4948 × cxp 5036 ^◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 fBascfbas 19555 Filcfil 21459 UnifOncust 21813

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-fbas 19564 df-fil 21460 df-ust 21814

This theorem is referenced by: (None)

W3C validator