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Theorem ustfn 22737
Description: The defined uniform structure as a function. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ustfn UnifOn Fn V

Proof of Theorem ustfn
Dummy variables 𝑣 𝑢 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 velpw 4543 . . . . 5 (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
21abbii 2883 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
3 abid2 2954 . . . . 5 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
4 vex 3495 . . . . . . . 8 𝑥 ∈ V
54, 4xpex 7465 . . . . . . 7 (𝑥 × 𝑥) ∈ V
65pwex 5272 . . . . . 6 𝒫 (𝑥 × 𝑥) ∈ V
76pwex 5272 . . . . 5 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
83, 7eqeltri 2906 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
92, 8eqeltrri 2907 . . 3 {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
10 simp1 1128 . . . 4 ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣))) → 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
1110ss2abi 4040 . . 3 {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ⊆ {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
129, 11ssexi 5217 . 2 {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ∈ V
13 df-ust 22736 . 2 UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})
1412, 13fnmpti 6484 1 UnifOn Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079  wcel 2105  {cab 2796  wral 3135  wrex 3136  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535   I cid 5452   × cxp 5546  ccnv 5547  cres 5550  ccom 5552   Fn wfn 6343  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-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-fun 6350  df-fn 6351  df-ust 22736
This theorem is referenced by:  ustn0  22756  elrnust  22760  ustbas  22763
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