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Theorem ustuni 24251
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 24231 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2 ustssxp 24229 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
32ralrimiva 3144 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4 pwssb 5106 . . 3 (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
53, 4sylibr 234 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6 elpwuni 5110 . . 3 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 = (𝑋 × 𝑋)))
76biimpa 476 . 2 (((𝑋 × 𝑋) ∈ 𝑈𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 = (𝑋 × 𝑋))
81, 5, 7syl2anc 584 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  wral 3059  wss 3963  𝒫 cpw 4605   cuni 4912   × cxp 5687  cfv 6563  UnifOncust 24224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ust 24225
This theorem is referenced by:  tususs  24295  cnflduss  25404
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