Theorem ustuni 24142

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 24123	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2		ustssxp 24121	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
3	2	ralrimiva 3125	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4		pwssb 5051	. . 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 5055	. . 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 584	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  𝒫 cpw 4549  ∪ cuni 4858   × cxp 5617  ‘cfv 6486  UnifOncust 24116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-res 5631  df-iota 6442  df-fun 6488  df-fv 6494  df-ust 24117

This theorem is referenced by:  tususs  24185  cnflduss  25284