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Theorem ustuni 23506
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 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))

Proof of Theorem ustuni
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ustbasel 23486 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2 ustssxp 23484 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
32ralrimiva 3142 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4 pwssb 5060 . . 3 (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
53, 4sylibr 233 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6 elpwuni 5064 . . 3 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 = (𝑋 × 𝑋)))
76biimpa 478 . 2 (((𝑋 × 𝑋) ∈ 𝑈𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 = (𝑋 × 𝑋))
81, 5, 7syl2anc 585 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wral 3063  wss 3909  𝒫 cpw 4559   cuni 4864   × cxp 5629  cfv 6492  UnifOncust 23479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6444  df-fun 6494  df-fv 6500  df-ust 23480
This theorem is referenced by:  tususs  23550  cnflduss  24648
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