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Theorem ustexhalf 22502
Description: For each entourage 𝑉 there is an entourage 𝑤 that is "not more than half as large". Condition UIII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
ustexhalf ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)
Distinct variable groups:   𝑤,𝑈   𝑤,𝑋   𝑤,𝑉

Proof of Theorem ustexhalf
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 elfvex 6571 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2 isust 22495 . . . . . . 7 (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 . . . . . 6 (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 . . . . 5 (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 1137 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
6 sseq1 3913 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
76imbi1d 343 . . . . . . 7 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
87ralbidv 3164 . . . . . 6 (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈)))
9 ineq1 4101 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑤) = (𝑉𝑤))
109eleq1d 2867 . . . . . . 7 (𝑣 = 𝑉 → ((𝑣𝑤) ∈ 𝑈 ↔ (𝑉𝑤) ∈ 𝑈))
1110ralbidv 3164 . . . . . 6 (𝑣 = 𝑉 → (∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ↔ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈))
12 sseq2 3914 . . . . . . 7 (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13 cnveq 5630 . . . . . . . 8 (𝑣 = 𝑉𝑣 = 𝑉)
1413eleq1d 2867 . . . . . . 7 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
15 sseq2 3914 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ 𝑉))
1615rexbidv 3260 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
1712, 14, 163anbi123d 1428 . . . . . 6 (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
188, 11, 173anbi123d 1428 . . . . 5 (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 3555 . . . 4 (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 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
2120simp3d 1137 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
2221simp3d 1137 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wrex 3106  Vcvv 3437  cin 3858  wss 3859  𝒫 cpw 4453   I cid 5347   × cxp 5441  ccnv 5442  cres 5445  ccom 5447  cfv 6225  UnifOncust 22491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-res 5455  df-iota 6189  df-fun 6227  df-fv 6233  df-ust 22492
This theorem is referenced by:  ustexsym  22507  ustex2sym  22508  ustex3sym  22509  trust  22521  ustuqtop4  22536  neipcfilu  22588
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