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Theorem ustuni 21940
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 21920 . 2 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2 ustssxp 21918 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
32ralrimiva 2960 . . 3 (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4 pwssb 4578 . . 3 (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢𝑈 𝑢 ⊆ (𝑋 × 𝑋))
53, 4sylibr 224 . 2 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6 elpwuni 4579 . . 3 ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ 𝑈 = (𝑋 × 𝑋)))
76biimpa 501 . 2 (((𝑋 × 𝑋) ∈ 𝑈𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → 𝑈 = (𝑋 × 𝑋))
81, 5, 7syl2anc 692 1 (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (𝑋 × 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2907  wss 3555  𝒫 cpw 4130   cuni 4402   × cxp 5072  cfv 5847  UnifOncust 21913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-res 5086  df-iota 5810  df-fun 5849  df-fv 5855  df-ust 21914
This theorem is referenced by:  tususs  21984  cnflduss  23060
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