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Theorem ustfn 22529
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 selpw 4424 . . . . 5 (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
21abbii 2839 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
3 abid2 2904 . . . . 5 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
4 vex 3413 . . . . . . . 8 𝑥 ∈ V
54, 4xpex 7292 . . . . . . 7 (𝑥 × 𝑥) ∈ V
65pwex 5131 . . . . . 6 𝒫 (𝑥 × 𝑥) ∈ V
76pwex 5131 . . . . 5 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
83, 7eqeltri 2857 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
92, 8eqeltrri 2858 . . 3 {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
10 simp1 1117 . . . 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 3928 . . 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 5079 . 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 22528 . 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 6319 1 UnifOn Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1069  wcel 2051  {cab 2753  wral 3083  wrex 3084  Vcvv 3410  cin 3823  wss 3824  𝒫 cpw 4417   I cid 5308   × cxp 5402  ccnv 5403  cres 5406  ccom 5408   Fn wfn 6181  UnifOncust 22527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-pw 4419  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-fun 6188  df-fn 6189  df-ust 22528
This theorem is referenced by:  ustn0  22548  elrnust  22552  ustbas  22555
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