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Theorem ustdiag 22232
Description: The diagonal set is included in any entourage, i.e. any point is 𝑉 -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
ustdiag ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)

Proof of Theorem ustdiag
Dummy variables 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6364 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 ∈ V)
2 isust 22227 . . . . . . 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 256 . . . . 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 1138 . . . 4 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑣𝑈 (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣)))
6 sseq1 3775 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑤𝑉𝑤))
76imbi1d 330 . . . . . . 7 (𝑣 = 𝑉 → ((𝑣𝑤𝑤𝑈) ↔ (𝑉𝑤𝑤𝑈)))
87ralbidv 3135 . . . . . 6 (𝑣 = 𝑉 → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑣𝑤𝑤𝑈) ↔ ∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈)))
9 ineq1 3958 . . . . . . . 8 (𝑣 = 𝑉 → (𝑣𝑤) = (𝑉𝑤))
109eleq1d 2835 . . . . . . 7 (𝑣 = 𝑉 → ((𝑣𝑤) ∈ 𝑈 ↔ (𝑉𝑤) ∈ 𝑈))
1110ralbidv 3135 . . . . . 6 (𝑣 = 𝑉 → (∀𝑤𝑈 (𝑣𝑤) ∈ 𝑈 ↔ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈))
12 sseq2 3776 . . . . . . 7 (𝑣 = 𝑉 → (( I ↾ 𝑋) ⊆ 𝑣 ↔ ( I ↾ 𝑋) ⊆ 𝑉))
13 cnveq 5433 . . . . . . . 8 (𝑣 = 𝑉𝑣 = 𝑉)
1413eleq1d 2835 . . . . . . 7 (𝑣 = 𝑉 → (𝑣𝑈𝑉𝑈))
15 sseq2 3776 . . . . . . . 8 (𝑣 = 𝑉 → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ 𝑉))
1615rexbidv 3200 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
1712, 14, 163anbi123d 1547 . . . . . 6 (𝑣 = 𝑉 → ((( I ↾ 𝑋) ⊆ 𝑣𝑣𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
188, 11, 173anbi123d 1547 . . . . 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 3456 . . . 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 496 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (∀𝑤 ∈ 𝒫 (𝑋 × 𝑋)(𝑉𝑤𝑤𝑈) ∧ ∀𝑤𝑈 (𝑉𝑤) ∈ 𝑈 ∧ (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉)))
2120simp3d 1138 . 2 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → (( I ↾ 𝑋) ⊆ 𝑉𝑉𝑈 ∧ ∃𝑤𝑈 (𝑤𝑤) ⊆ 𝑉))
2221simp1d 1136 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → ( I ↾ 𝑋) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  cin 3722  wss 3723  𝒫 cpw 4298   I cid 5157   × cxp 5248  ccnv 5249  cres 5252  ccom 5254  cfv 6030  UnifOncust 22223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-res 5262  df-iota 5993  df-fun 6032  df-fv 6038  df-ust 22224
This theorem is referenced by:  ustssco  22238  ustref  22242  ustelimasn  22246  trust  22253  ustuqtop3  22267
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