MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustuni Structured version   Visualization version   GIF version
Theorem ustuni 23730
Description: The set union of a uniform structure is the Cartesian product of its base. (Contributed by Thierry Arnoux, 5-Dec-2017.)
Assertion
Ref Expression
ustuni (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
Proof of Theorem ustuni
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ustbasel 23710 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑋 Γ— 𝑋) ∈ π‘ˆ)
2 ustssxp 23708 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑒 ∈ π‘ˆ) β†’ 𝑒 βŠ† (𝑋 Γ— 𝑋))
32ralrimiva 3146 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
4 pwssb 5104 . . 3 (π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋) ↔ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
53, 4sylibr 233 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋))
6 elpwuni 5108 . . 3 ((𝑋 Γ— 𝑋) ∈ π‘ˆ β†’ (π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋) ↔ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋)))
76biimpa 477 . 2 (((𝑋 Γ— 𝑋) ∈ π‘ˆ ∧ π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋)) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
81, 5, 7syl2anc 584 1 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948  π’« cpw 4602  βˆͺ cuni 4908   Γ— cxp 5674  β€˜cfv 6543  UnifOncust 23703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551  df-ust 23704
This theorem is referenced by:  tususs  23774  cnflduss  24872
  Copyright terms: Public domain W3C validator