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Theorem ustincl 22750
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 6702 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2 isust 22746 . . . . . . . 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 268 . . . . . 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 1138 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
6 sseq1 3996 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
76imbi1d 343 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
87ralbidv 3202 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈)))
9 ineq1 4185 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤) = (𝑉𝑤))
109eleq1d 2902 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤) ∈ 𝑈 ↔ (𝑉𝑤) ∈ 𝑈))
1110ralbidv 3202 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ↔ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈))
12 sseq2 3997 . . . . . . . 8 (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13 cnveq 5743 . . . . . . . . 9 (𝑣 = 𝑉𝑣 = 𝑉)
1413eleq1d 2902 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
15 sseq2 3997 . . . . . . . . 9 (𝑣 = 𝑉 → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ 𝑉))
1615rexbidv 3302 . . . . . . . 8 (𝑣 = 𝑉 → (∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
1712, 14, 163anbi123d 1429 . . . . . . 7 (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
188, 11, 173anbi123d 1429 . . . . . 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 3622 . . . . 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 507 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
2120simp2d 1137 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈)
22 ineq2 4187 . . . . 5 (𝑤 = 𝑊 → (𝑉𝑤) = (𝑉𝑊))
2322eleq1d 2902 . . . 4 (𝑤 = 𝑊 → ((𝑉𝑤) ∈ 𝑈 ↔ (𝑉𝑊) ∈ 𝑈))
2423rspcv 3622 . . 3 (𝑊𝑈 → (∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 → (𝑉𝑊) ∈ 𝑈))
2521, 24mpan9 507 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
26253impa 1104 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wrex 3144  Vcvv 3500  cin 3939  wss 3940  𝒫 cpw 4542   I cid 5458   × cxp 5552  ccnv 5553  cres 5556  ccom 5558  cfv 6354  UnifOncust 22742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-res 5566  df-iota 6313  df-fun 6356  df-fv 6362  df-ust 22743
This theorem is referenced by:  ustexsym  22758  trust  22772  utoptop  22777  restutopopn  22781  ustuqtop2  22785
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