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Theorem ustssel 24214
Description: A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.) (Proof shortened by AV, 17-Sep-2021.)
Assertion
Ref Expression
ustssel ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉𝑊𝑊𝑈))

Proof of Theorem ustssel
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
21elfvexd 6945 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑋 ∈ V)
3 isust 24212 . . . . . 6 (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)))))
42, 3syl 17 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 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)))))
51, 4mpbid 232 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ∈ 𝑈 ∧ ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣))))
65simp3d 1145 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
7 simp1 1137 . . . 4 ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈))
87ralimi 3083 . . 3 (∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)) → ∀𝑣𝑈𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈))
96, 8syl 17 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → ∀𝑣𝑈𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈))
10 simp2 1138 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
112, 2xpexd 7771 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) ∈ V)
12 simp3 1139 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑊 ⊆ (𝑋 × 𝑋))
1311, 12sselpwd 5328 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑊 ∈ 𝒫 (𝑋 × 𝑋))
14 sseq1 4009 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
1514imbi1d 341 . . . 4 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
16 sseq2 4010 . . . . 5 (𝑤 = 𝑊 → (𝑉𝑤𝑉𝑊))
17 eleq1 2829 . . . . 5 (𝑤 = 𝑊 → (𝑤𝑈𝑊𝑈))
1816, 17imbi12d 344 . . . 4 (𝑤 = 𝑊 → ((𝑉𝑤𝑤𝑈) ↔ (𝑉𝑊𝑊𝑈)))
1915, 18rspc2v 3633 . . 3 ((𝑉𝑈𝑊 ∈ 𝒫 (𝑋 × 𝑋)) → (∀𝑣𝑈𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) → (𝑉𝑊𝑊𝑈)))
2010, 13, 19syl2anc 584 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (∀𝑣𝑈𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) → (𝑉𝑊𝑊𝑈)))
219, 20mpd 15 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑉𝑊𝑊𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  cin 3950  wss 3951  𝒫 cpw 4600   I cid 5577   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  cfv 6561  UnifOncust 24208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ust 24209
This theorem is referenced by:  trust  24238  ustuqtop1  24250  ucnprima  24291
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