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Theorem ustuni 24075
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 24055 . 2 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ (𝑋 Γ— 𝑋) ∈ π‘ˆ)
2 ustssxp 24053 . . . 4 ((π‘ˆ ∈ (UnifOnβ€˜π‘‹) ∧ 𝑒 ∈ π‘ˆ) β†’ 𝑒 βŠ† (𝑋 Γ— 𝑋))
32ralrimiva 3138 . . 3 (π‘ˆ ∈ (UnifOnβ€˜π‘‹) β†’ βˆ€π‘’ ∈ π‘ˆ 𝑒 βŠ† (𝑋 Γ— 𝑋))
4 pwssb 5095 . . 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 5099 . . 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 1533   ∈ wcel 2098  βˆ€wral 3053   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900   Γ— cxp 5665  β€˜cfv 6534  UnifOncust 24048
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-res 5679  df-iota 6486  df-fun 6536  df-fv 6542  df-ust 24049
This theorem is referenced by:  tususs  24119  cnflduss  25228
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