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Theorem ustund 22258
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 5902 . . 3 ((𝐴𝐵) ≠ ∅ → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
31, 2syl 17 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) = ((𝐴𝐵) × (𝐴𝐵)))
4 xpundir 5385 . . . . 5 ((𝐴𝐵) × (𝐴𝐵)) = ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵)))
5 xpindi 5470 . . . . . . 7 (𝐴 × (𝐴𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
6 inss1 4040 . . . . . . . 8 ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
7 ustund.1 . . . . . . . 8 (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
86, 7syl5ss 3820 . . . . . . 7 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
95, 8syl5eqss 3857 . . . . . 6 (𝜑 → (𝐴 × (𝐴𝐵)) ⊆ 𝑉)
10 xpindi 5470 . . . . . . 7 (𝐵 × (𝐴𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
11 inss2 4041 . . . . . . . 8 ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12 ustund.2 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
1311, 12syl5ss 3820 . . . . . . 7 (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
1410, 13syl5eqss 3857 . . . . . 6 (𝜑 → (𝐵 × (𝐴𝐵)) ⊆ 𝑉)
159, 14unssd 3999 . . . . 5 (𝜑 → ((𝐴 × (𝐴𝐵)) ∪ (𝐵 × (𝐴𝐵))) ⊆ 𝑉)
164, 15syl5eqss 3857 . . . 4 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
17 coss2 5493 . . . 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 5384 . . . . 5 ((𝐴𝐵) × (𝐴𝐵)) = (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵))
20 xpindir 5471 . . . . . . 7 ((𝐴𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
21 inss1 4040 . . . . . . . 8 ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
2221, 7syl5ss 3820 . . . . . . 7 (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
2320, 22syl5eqss 3857 . . . . . 6 (𝜑 → ((𝐴𝐵) × 𝐴) ⊆ 𝑉)
24 xpindir 5471 . . . . . . 7 ((𝐴𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
25 inss2 4041 . . . . . . . 8 ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
2625, 12syl5ss 3820 . . . . . . 7 (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
2724, 26syl5eqss 3857 . . . . . 6 (𝜑 → ((𝐴𝐵) × 𝐵) ⊆ 𝑉)
2823, 27unssd 3999 . . . . 5 (𝜑 → (((𝐴𝐵) × 𝐴) ∪ ((𝐴𝐵) × 𝐵)) ⊆ 𝑉)
2919, 28syl5eqss 3857 . . . 4 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉)
30 coss1 5492 . . . 4 (((𝐴𝐵) × (𝐴𝐵)) ⊆ 𝑉 → (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉) ⊆ (𝑉𝑉))
3129, 30syl 17 . . 3 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ 𝑉) ⊆ (𝑉𝑉))
3218, 31sstrd 3819 . 2 (𝜑 → (((𝐴𝐵) × (𝐴𝐵)) ∘ ((𝐴𝐵) × (𝐴𝐵))) ⊆ (𝑉𝑉))
333, 32eqsstr3d 3848 1 (𝜑 → ((𝐴𝐵) × (𝐴𝐵)) ⊆ (𝑉𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wne 2989  cun 3778  cin 3779  wss 3780  c0 4127   × cxp 5322  ccom 5328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-br 4856  df-opab 4918  df-xp 5330  df-rel 5331  df-co 5333
This theorem is referenced by: (None)
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