Theorem ustuni 24182

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.

Step	Hyp	Ref	Expression
1		ustbasel 24163	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2		ustssxp 24161	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
3	2	ralrimiva 3130	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4		pwssb 5058	. . 3 ⊢ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋))
5	3, 4	sylibr 234	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6		elpwuni 5062	. . 3 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∪ 𝑈 = (𝑋 × 𝑋)))
7	6	biimpa 476	. 2 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → ∪ 𝑈 = (𝑋 × 𝑋))
8	1, 5, 7	syl2anc 585	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865   × cxp 5630  ‘cfv 6500  UnifOncust 24156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fv 6508  df-ust 24157

This theorem is referenced by:  tususs  24225  cnflduss  25324