Theorem ustund 24574

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 → (𝐴 ∩ 𝐵) ≠ ∅)

Assertion

Ref	Expression
ustund	⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))

Proof of Theorem ustund

Step	Hyp	Ref	Expression
1		ustund.3	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ≠ ∅)
2		xpco 6313	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) = ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)))
3	1, 2	syl 18	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) = ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)))
4		xpundi 5725	. . . 4 ⊢ ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) = (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵))
5		xpindir 5818	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
6		inss1 4136	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
7		ustund.1	. . . . . . 7 ⊢ (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
8	6, 7	sstrid 3924	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
9	5, 8	eqsstrid 3951	. . . . 5 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐴) ⊆ 𝑉)
10		xpindir 5818	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
11		inss2 4137	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12		ustund.2	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
13	11, 12	sstrid 3924	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
14	10, 13	eqsstrid 3951	. . . . 5 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐵) ⊆ 𝑉)
15	9, 14	unssd 4193	. . . 4 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵)) ⊆ 𝑉)
16	4, 15	eqsstrid 3951	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ 𝑉)
17		xpundir 5726	. . . 4 ⊢ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) = ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵)))
18		xpindi 5817	. . . . . 6 ⊢ (𝐴 × (𝐴 ∩ 𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
19		inss1 4136	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
20	19, 7	sstrid 3924	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
21	18, 20	eqsstrid 3951	. . . . 5 ⊢ (𝜑 → (𝐴 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
22		xpindi 5817	. . . . . 6 ⊢ (𝐵 × (𝐴 ∩ 𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
23		inss2 4137	. . . . . . 7 ⊢ ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
24	23, 12	sstrid 3924	. . . . . 6 ⊢ (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
25	22, 24	eqsstrid 3951	. . . . 5 ⊢ (𝜑 → (𝐵 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
26	21, 25	unssd 4193	. . . 4 ⊢ (𝜑 → ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵))) ⊆ 𝑉)
27	17, 26	eqsstrid 3951	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
28	16, 27	coss12d 15157	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (𝑉 ∘ 𝑉))
29	3, 28	eqsstrrd 3948	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2959   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   × cxp 5654   ∘ ccom 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-xp 5662  df-rel 5663  df-co 5665

This theorem is used by: (None)