Theorem ustincl 24268

Description: A uniform structure is closed under finite intersection. Condition F_II of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)

Assertion

Ref	Expression
ustincl	⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ∈ 𝑈) → (𝑉 ∩ 𝑊) ∈ 𝑈)

Proof of Theorem ustincl

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

Step	Hyp	Ref	Expression
1		elfvex 6902	. . . . . . . 8 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2		isust 24264	. . . . . . . 8 ⊢ (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	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (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	simp3d 1157	. . . . 5 ⊢ (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)))
6		sseq1 3961	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ 𝑤 ↔ 𝑉 ⊆ 𝑤))
7	6	imbi1d 343	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ↔ (𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈)))
8	7	ralbidv 3185	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈)))
9		ineq1 4165	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → (𝑣 ∩ 𝑤) = (𝑉 ∩ 𝑤))
10	9	eleq1d 2847	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → ((𝑣 ∩ 𝑤) ∈ 𝑈 ↔ (𝑉 ∩ 𝑤) ∈ 𝑈))
11	10	ralbidv 3185	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ↔ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈))
12		sseq2 3962	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13		cnveq 5845	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → ◡𝑣 = ◡𝑉)
14	13	eleq1d 2847	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (◡𝑣 ∈ 𝑈 ↔ ◡𝑉 ∈ 𝑈))
15		sseq2 3962	. . . . . . . . 9 ⊢ (𝑣 = 𝑉 → ((𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ (𝑤 ∘ 𝑤) ⊆ 𝑉))
16	15	rexbidv 3186	. . . . . . . 8 ⊢ (𝑣 = 𝑉 → (∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))
17	12, 14, 16	3anbi123d 1457	. . . . . . 7 ⊢ (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉)))
18	8, 11, 17	3anbi123d 1457	. . . . . 6 ⊢ (𝑣 = 𝑉 → ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑣 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))))
19	18	rspcv 3577	. . . . 5 ⊢ (𝑉 ∈ 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)) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉 ⊆ 𝑤 → 𝑤 ∈ 𝑈) ∧ ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉 ∧ ◡𝑉 ∈ 𝑈 ∧ ∃𝑤 ∈ 𝑈 (𝑤 ∘ 𝑤) ⊆ 𝑉))))
20	5, 19	mpan9 514	. . . 4 ⊢ ((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)))
21	20	simp2d 1156	. . 3 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → ∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈)
22		ineq2 4166	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑉 ∩ 𝑤) = (𝑉 ∩ 𝑊))
23	22	eleq1d 2847	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑉 ∩ 𝑤) ∈ 𝑈 ↔ (𝑉 ∩ 𝑊) ∈ 𝑈))
24	23	rspcv 3577	. . 3 ⊢ (𝑊 ∈ 𝑈 → (∀𝑤 ∈ 𝑈 (𝑉 ∩ 𝑤) ∈ 𝑈 → (𝑉 ∩ 𝑊) ∈ 𝑈))
25	21, 24	mpan9 514	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) ∧ 𝑊 ∈ 𝑈) → (𝑉 ∩ 𝑊) ∈ 𝑈)
26	25	3impa 1122	1 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑊 ∈ 𝑈) → (𝑉 ∩ 𝑊) ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4555   I cid 5541   × cxp 5645  ^◡ccnv 5646   ↾ cres 5649   ∘ ccom 5651  ‘cfv 6521  UnifOncust 24260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-iota 6477  df-fun 6523  df-fv 6529  df-ust 24261

This theorem is referenced by:  ustexsym  24276  trust  24289  utoptop  24294  restutopopn  24298  ustuqtop2  24302