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Theorem ustund 23946
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 6288 . . 3 ((𝐴𝐵) ≠ ∅ → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
31, 2syl 17 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
4 xpundi 5744 . . . 4 ((𝐴𝐵) × (𝐴𝐵)) = (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵))
5 xpindir 5834 . . . . . 6 ((𝐴𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
6 inss1 4228 . . . . . . 7 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
7 ustund.1 . . . . . . 7 (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
86, 7sstrid 3993 . . . . . 6 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
95, 8eqsstrid 4030 . . . . 5 (𝜑 → ((𝐴𝐵) × 𝐴) ⊆ 𝑉)
10 xpindir 5834 . . . . . 6 ((𝐴𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
11 inss2 4229 . . . . . . 7 ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12 ustund.2 . . . . . . 7 (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
1311, 12sstrid 3993 . . . . . 6 (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
1410, 13eqsstrid 4030 . . . . 5 (𝜑 → ((𝐴𝐵) × 𝐵) ⊆ 𝑉)
159, 14unssd 4186 . . . 4 (𝜑 → (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵)) ⊆ 𝑉)
164, 15eqsstrid 4030 . . 3 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
17 xpundir 5745 . . . 4 ((𝐴𝐵) × (𝐴𝐵)) = ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵)))
18 xpindi 5833 . . . . . 6 (𝐴 × (𝐴𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
19 inss1 4228 . . . . . . 7 ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
2019, 7sstrid 3993 . . . . . 6 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
2118, 20eqsstrid 4030 . . . . 5 (𝜑 → (𝐴 × (𝐴𝐵)) ⊆ 𝑉)
22 xpindi 5833 . . . . . 6 (𝐵 × (𝐴𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
23 inss2 4229 . . . . . . 7 ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2423, 12sstrid 3993 . . . . . 6 (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
2522, 24eqsstrid 4030 . . . . 5 (𝜑 → (𝐵 × (𝐴𝐵)) ⊆ 𝑉)
2621, 25unssd 4186 . . . 4 (𝜑 → ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵))) ⊆ 𝑉)
2717, 26eqsstrid 4030 . . 3 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
2816, 27coss12d 14923 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (𝑉𝑉))
293, 28eqsstrrd 4021 1 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wne 2940  cun 3946  cin 3947  wss 3948  c0 4322   × cxp 5674  ccom 5680
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-co 5685
This theorem is referenced by: (None)
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