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Theorem ustincl 23110
Description: A uniform structure is closed under finite intersection. Condition FII of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 30-Nov-2017.)
Assertion
Ref Expression
ustincl ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)

Proof of Theorem ustincl
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6755 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2 isust 23106 . . . . . . . 8 (𝑋 ∈ V → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
31, 2syl 17 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ∈ (UnifOn‘𝑋) ↔ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))))
43ibi 270 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))))
54simp3d 1146 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
6 sseq1 3931 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
76imbi1d 345 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
87ralbidv 3118 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈)))
9 ineq1 4125 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤) = (𝑉𝑤))
109eleq1d 2822 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤) ∈ 𝑈 ↔ (𝑉𝑤) ∈ 𝑈))
1110ralbidv 3118 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ↔ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈))
12 sseq2 3932 . . . . . . . 8 (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13 cnveq 5747 . . . . . . . . 9 (𝑣 = 𝑉𝑣 = 𝑉)
1413eleq1d 2822 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
15 sseq2 3932 . . . . . . . . 9 (𝑣 = 𝑉 → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ 𝑉))
1615rexbidv 3221 . . . . . . . 8 (𝑣 = 𝑉 → (∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
1712, 14, 163anbi123d 1438 . . . . . . 7 (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
188, 11, 173anbi123d 1438 . . . . . 6 (𝑣 = 𝑉 → ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))))
1918rspcv 3537 . . . . 5 (𝑉𝑈 → (∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))))
205, 19mpan9 510 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
2120simp2d 1145 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈)
22 ineq2 4126 . . . . 5 (𝑤 = 𝑊 → (𝑉𝑤) = (𝑉𝑊))
2322eleq1d 2822 . . . 4 (𝑤 = 𝑊 → ((𝑉𝑤) ∈ 𝑈 ↔ (𝑉𝑊) ∈ 𝑈))
2423rspcv 3537 . . 3 (𝑊𝑈 → (∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 → (𝑉𝑊) ∈ 𝑈))
2521, 24mpan9 510 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
26253impa 1112 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3413  cin 3870  wss 3871  𝒫 cpw 4518   I cid 5459   × cxp 5554  ccnv 5555  cres 5558  ccom 5560  cfv 6385  UnifOncust 23102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-br 5059  df-opab 5121  df-mpt 5141  df-id 5460  df-xp 5562  df-rel 5563  df-cnv 5564  df-co 5565  df-dm 5566  df-res 5568  df-iota 6343  df-fun 6387  df-fv 6393  df-ust 23103
This theorem is referenced by:  ustexsym  23118  trust  23132  utoptop  23137  restutopopn  23141  ustuqtop2  23145
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