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Theorem ustincl 21758
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 6111 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2 isust 21754 . . . . . . . 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 254 . . . . . 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 1067 . . . . 5 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
6 sseq1 3583 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
76imbi1d 329 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
87ralbidv 2963 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈)))
9 ineq1 3763 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑣𝑤) = (𝑉𝑤))
109eleq1d 2666 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑣𝑤) ∈ 𝑈 ↔ (𝑉𝑤) ∈ 𝑈))
1110ralbidv 2963 . . . . . . 7 (𝑣 = 𝑉 → (∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ↔ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈))
12 sseq2 3584 . . . . . . . 8 (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13 cnveq 5201 . . . . . . . . 9 (𝑣 = 𝑉𝑣 = 𝑉)
1413eleq1d 2666 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
15 sseq2 3584 . . . . . . . . 9 (𝑣 = 𝑉 → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ 𝑉))
1615rexbidv 3028 . . . . . . . 8 (𝑣 = 𝑉 → (∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
1712, 14, 163anbi123d 1390 . . . . . . 7 (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
188, 11, 173anbi123d 1390 . . . . . 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 3272 . . . . 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 484 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
2120simp2d 1066 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈)
22 ineq2 3764 . . . . 5 (𝑤 = 𝑊 → (𝑉𝑤) = (𝑉𝑊))
2322eleq1d 2666 . . . 4 (𝑤 = 𝑊 → ((𝑉𝑤) ∈ 𝑈 ↔ (𝑉𝑊) ∈ 𝑈))
2423rspcv 3272 . . 3 (𝑊𝑈 → (∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 → (𝑉𝑊) ∈ 𝑈))
2521, 24mpan9 484 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) ∧ 𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
26253impa 1250 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊𝑈) → (𝑉𝑊) ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2890  wrex 2891  Vcvv 3167  cin 3533  wss 3534  𝒫 cpw 4102   I cid 4933   × cxp 5021  ccnv 5022  cres 5025  ccom 5027  cfv 5785  UnifOncust 21750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-res 5035  df-iota 5749  df-fun 5787  df-fv 5793  df-ust 21751
This theorem is referenced by:  ustexsym  21766  trust  21780  utoptop  21785  restutopopn  21789  ustuqtop2  21793
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