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Theorem ustuni 22249
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 22229 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2 ustssxp 22227 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
32ralrimiva 3114 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4 pwssb 4744 . . 3 (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
53, 4sylibr 224 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6 elpwuni 4748 . . 3 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 = (𝑋 × 𝑋)))
76biimpa 462 . 2 (((𝑋 × 𝑋) ∈ 𝑈𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 = (𝑋 × 𝑋))
81, 5, 7syl2anc 565 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  wral 3060  wss 3721  𝒫 cpw 4295   cuni 4572   × cxp 5247  cfv 6031  UnifOncust 22222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ust 22223
This theorem is referenced by:  tususs  22293  cnflduss  23370
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