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Theorem ustssel 22741
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 1128 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
21elfvexd 6697 . . . . . 6 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑋 ∈ V)
3 isust 22739 . . . . . 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 233 . . . 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 1136 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
7 simp1 1128 . . . 4 ((∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)) → ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈))
87ralimi 3157 . . 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 1129 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
112, 2xpexd 7463 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) ∈ V)
12 simp3 1130 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑊 ⊆ (𝑋 × 𝑋))
1311, 12sselpwd 5221 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑊 ⊆ (𝑋 × 𝑋)) → 𝑊 ∈ 𝒫 (𝑋 × 𝑋))
14 sseq1 3989 . . . . 5 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
1514imbi1d 343 . . . 4 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
16 sseq2 3990 . . . . 5 (𝑤 = 𝑊 → (𝑉𝑤𝑉𝑊))
17 eleq1 2897 . . . . 5 (𝑤 = 𝑊 → (𝑤𝑈𝑊𝑈))
1816, 17imbi12d 346 . . . 4 (𝑤 = 𝑊 → ((𝑉𝑤𝑤𝑈) ↔ (𝑉𝑊𝑊𝑈)))
1915, 18rspc2v 3630 . . 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 207  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535   I cid 5452   × cxp 5546  ccnv 5547  cres 5550  ccom 5552  cfv 6348  UnifOncust 22735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-ust 22736
This theorem is referenced by:  trust  22765  ustuqtop1  22777  ucnprima  22818
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