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Theorem ustuni 24144
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 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
Proof of Theorem ustuni
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 ustbasel 24124 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑋 Γ— 𝑋) ∈ π‘ˆ)
2 ustssxp 24122 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑒 ∈ π‘ˆ) β†’ 𝑒 βŠ† (𝑋 Γ— 𝑋))
32ralrimiva 3143 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
4 pwssb 5104 . . 3 (π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋) ↔ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
53, 4sylibr 233 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋))
6 elpwuni 5108 . . 3 ((𝑋 Γ— 𝑋) ∈ π‘ˆ β†’ (π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋) ↔ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋)))
76biimpa 476 . 2 (((𝑋 Γ— 𝑋) ∈ π‘ˆ ∧ π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋)) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
81, 5, 7syl2anc 583 1 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βŠ† wss 3947  π’« cpw 4603  βˆͺ cuni 4908   Γ— cxp 5676  β€˜cfv 6548  UnifOncust 24117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-ust 24118
This theorem is referenced by:  tususs  24188  cnflduss  25297
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