Theorem ustuni 24216

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 24197	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) ∈ 𝑈)
2		ustssxp 24195	. . . 4 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑢 ∈ 𝑈) → 𝑢 ⊆ (𝑋 × 𝑋))
3	2	ralrimiva 3132	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋))
4		pwssb 5037	. . 3 ⊢ (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∀𝑢 ∈ 𝑈 𝑢 ⊆ (𝑋 × 𝑋))
5	3, 4	sylibr 235	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 ⊆ 𝒫 (𝑋 × 𝑋))
6		elpwuni 5041	. . 3 ⊢ ((𝑋 × 𝑋) ∈ 𝑈 → (𝑈 ⊆ 𝒫 (𝑋 × 𝑋) ↔ ∪ 𝑈 = (𝑋 × 𝑋)))
7	6	biimpa 477	. 2 ⊢ (((𝑋 × 𝑋) ∈ 𝑈 ∧ 𝑈 ⊆ 𝒫 (𝑋 × 𝑋)) → ∪ 𝑈 = (𝑋 × 𝑋))
8	1, 5, 7	syl2anc 590	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 = (𝑋 × 𝑋))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3054   ⊆ wss 3890  𝒫 cpw 4536  ∪ cuni 4845   × cxp 5623  ‘cfv 6492  UnifOncust 24190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-res 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ust 24191

This theorem is referenced by:  tususs  24259  cnflduss  25348