MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustfn Structured version   Visualization version   GIF version

Theorem ustfn 24211
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 4604 . . . . 5 (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
21abbii 2808 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
3 abid2 2878 . . . . 5 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
4 vex 3483 . . . . . . . 8 𝑥 ∈ V
54, 4xpex 7774 . . . . . . 7 (𝑥 × 𝑥) ∈ V
65pwex 5379 . . . . . 6 𝒫 (𝑥 × 𝑥) ∈ V
76pwex 5379 . . . . 5 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
83, 7eqeltri 2836 . . . 4 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
92, 8eqeltrri 2837 . . 3 {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
10 simp1 1136 . . . 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 4066 . . 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 5321 . 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 24210 . 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 6710 1 UnifOn Fn V
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  𝒫 cpw 4599   I cid 5576   × cxp 5682  ccnv 5683  cres 5686  ccom 5688   Fn wfn 6555  UnifOncust 24209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-fun 6562  df-fn 6563  df-ust 24210
This theorem is referenced by:  ustn0  24230  ustbas  24237
  Copyright terms: Public domain W3C validator