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Theorem ustuni 24256
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 24236 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2 ustssxp 24234 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
32ralrimiva 3152 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4 pwssb 5124 . . 3 (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
53, 4sylibr 234 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6 elpwuni 5128 . . 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 583 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  wral 3067  wss 3976  𝒫 cpw 4622   cuni 4931   × cxp 5698  cfv 6573  UnifOncust 24229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-ust 24230
This theorem is referenced by:  tususs  24300  cnflduss  25409
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