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Theorem ustuni 23594
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 23574 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑋 Γ— 𝑋) ∈ π‘ˆ)
2 ustssxp 23572 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑒 ∈ π‘ˆ) β†’ 𝑒 βŠ† (𝑋 Γ— 𝑋))
32ralrimiva 3140 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
4 pwssb 5062 . . 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 5066 . . 3 ((𝑋 Γ— 𝑋) ∈ π‘ˆ β†’ (π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋) ↔ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋)))
76biimpa 478 . 2 (((𝑋 Γ— 𝑋) ∈ π‘ˆ ∧ π‘ˆ βŠ† 𝒫 (𝑋 Γ— 𝑋)) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
81, 5, 7syl2anc 585 1 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆͺ π‘ˆ = (𝑋 Γ— 𝑋))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  π’« cpw 4561  βˆͺ cuni 4866   Γ— cxp 5632  β€˜cfv 6497  UnifOncust 23567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-iota 6449  df-fun 6499  df-fv 6505  df-ust 23568
This theorem is referenced by:  tususs  23638  cnflduss  24736
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