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Theorem ustund 22246
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 5836 . . 3 ((𝐴𝐵) ≠ ∅ → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
31, 2syl 17 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
4 xpundir 5329 . . . . 5 ((𝐴𝐵) × (𝐴𝐵)) = ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵)))
5 xpindi 5411 . . . . . . 7 (𝐴 × (𝐴𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
6 inss1 3976 . . . . . . . 8 ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
7 ustund.1 . . . . . . . 8 (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
86, 7syl5ss 3755 . . . . . . 7 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
95, 8syl5eqss 3790 . . . . . 6 (𝜑 → (𝐴 × (𝐴𝐵)) ⊆ 𝑉)
10 xpindi 5411 . . . . . . 7 (𝐵 × (𝐴𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
11 inss2 3977 . . . . . . . 8 ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12 ustund.2 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
1311, 12syl5ss 3755 . . . . . . 7 (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
1410, 13syl5eqss 3790 . . . . . 6 (𝜑 → (𝐵 × (𝐴𝐵)) ⊆ 𝑉)
159, 14unssd 3932 . . . . 5 (𝜑 → ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵))) ⊆ 𝑉)
164, 15syl5eqss 3790 . . . 4 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
17 coss2 5434 . . . 4 (((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉))
1816, 17syl 17 . . 3 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉))
19 xpundi 5328 . . . . 5 ((𝐴𝐵) × (𝐴𝐵)) = (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵))
20 xpindir 5412 . . . . . . 7 ((𝐴𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
21 inss1 3976 . . . . . . . 8 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
2221, 7syl5ss 3755 . . . . . . 7 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
2320, 22syl5eqss 3790 . . . . . 6 (𝜑 → ((𝐴𝐵) × 𝐴) ⊆ 𝑉)
24 xpindir 5412 . . . . . . 7 ((𝐴𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
25 inss2 3977 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2625, 12syl5ss 3755 . . . . . . 7 (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
2724, 26syl5eqss 3790 . . . . . 6 (𝜑 → ((𝐴𝐵) × 𝐵) ⊆ 𝑉)
2823, 27unssd 3932 . . . . 5 (𝜑 → (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵)) ⊆ 𝑉)
2919, 28syl5eqss 3790 . . . 4 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
30 coss1 5433 . . . 4 (((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉 → (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉) ⊆ (𝑉𝑉))
3129, 30syl 17 . . 3 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉) ⊆ (𝑉𝑉))
3218, 31sstrd 3754 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (𝑉𝑉))
333, 32eqsstr3d 3781 1 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wne 2932  cun 3713  cin 3714  wss 3715  c0 4058   × cxp 5264  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-co 5275
This theorem is referenced by: (None)
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