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Theorem ustund 22764
 Description: If two intersecting sets 𝐴 and 𝐵 are both small in 𝑉, their union is small in (𝑉↑2). Proposition 1 of [BourbakiTop1] p. II.12. This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 17-Nov-2017.)
Hypotheses
Ref Expression
ustund.1 (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
ustund.2 (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
ustund.3 (𝜑 → (𝐴𝐵) ≠ ∅)
Assertion
Ref Expression
ustund (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))

Proof of Theorem ustund
StepHypRef Expression
1 ustund.3 . . 3 (𝜑 → (𝐴𝐵) ≠ ∅)
2 xpco 6139 . . 3 ((𝐴𝐵) ≠ ∅ → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
31, 2syl 17 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
4 xpundi 5619 . . . 4 ((𝐴𝐵) × (𝐴𝐵)) = (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵))
5 xpindir 5704 . . . . . 6 ((𝐴𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
6 inss1 4209 . . . . . . 7 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
7 ustund.1 . . . . . . 7 (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
86, 7sstrid 3982 . . . . . 6 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
95, 8eqsstrid 4019 . . . . 5 (𝜑 → ((𝐴𝐵) × 𝐴) ⊆ 𝑉)
10 xpindir 5704 . . . . . 6 ((𝐴𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
11 inss2 4210 . . . . . . 7 ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12 ustund.2 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
1311, 12sstrid 3982 . . . . . 6 (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
1410, 13eqsstrid 4019 . . . . 5 (𝜑 → ((𝐴𝐵) × 𝐵) ⊆ 𝑉)
159, 14unssd 4166 . . . 4 (𝜑 → (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵)) ⊆ 𝑉)
164, 15eqsstrid 4019 . . 3 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
17 xpundir 5620 . . . 4 ((𝐴𝐵) × (𝐴𝐵)) = ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵)))
18 xpindi 5703 . . . . . 6 (𝐴 × (𝐴𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
19 inss1 4209 . . . . . . 7 ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
2019, 7sstrid 3982 . . . . . 6 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
2118, 20eqsstrid 4019 . . . . 5 (𝜑 → (𝐴 × (𝐴𝐵)) ⊆ 𝑉)
22 xpindi 5703 . . . . . 6 (𝐵 × (𝐴𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
23 inss2 4210 . . . . . . 7 ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2423, 12sstrid 3982 . . . . . 6 (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
2522, 24eqsstrid 4019 . . . . 5 (𝜑 → (𝐵 × (𝐴𝐵)) ⊆ 𝑉)
2621, 25unssd 4166 . . . 4 (𝜑 → ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵))) ⊆ 𝑉)
2717, 26eqsstrid 4019 . . 3 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
2816, 27coss12d 14327 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (𝑉𝑉))
293, 28eqsstrrd 4010 1 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ≠ wne 3021   ∪ cun 3938   ∩ cin 3939   ⊆ wss 3940  ∅c0 4295   × cxp 5552   ∘ ccom 5558 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-rel 5561  df-co 5563 This theorem is referenced by: (None)
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