Theorem ustund 24187

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 6253	. . 3 ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) = ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) = ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)))
4		xpundi 5700	. . . 4 ⊢ ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) = (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵))
5		xpindir 5789	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
6		inss1 4177	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
7		ustund.1	. . . . . . 7 ⊢ (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
8	6, 7	sstrid 3933	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
9	5, 8	eqsstrid 3960	. . . . 5 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐴) ⊆ 𝑉)
10		xpindir 5789	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
11		inss2 4178	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12		ustund.2	. . . . . . 7 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
13	11, 12	sstrid 3933	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
14	10, 13	eqsstrid 3960	. . . . 5 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐵) ⊆ 𝑉)
15	9, 14	unssd 4132	. . . 4 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵)) ⊆ 𝑉)
16	4, 15	eqsstrid 3960	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ 𝑉)
17		xpundir 5701	. . . 4 ⊢ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) = ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵)))
18		xpindi 5788	. . . . . 6 ⊢ (𝐴 × (𝐴 ∩ 𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
19		inss1 4177	. . . . . . 7 ⊢ ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
20	19, 7	sstrid 3933	. . . . . 6 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
21	18, 20	eqsstrid 3960	. . . . 5 ⊢ (𝜑 → (𝐴 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
22		xpindi 5788	. . . . . 6 ⊢ (𝐵 × (𝐴 ∩ 𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
23		inss2 4178	. . . . . . 7 ⊢ ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
24	23, 12	sstrid 3933	. . . . . 6 ⊢ (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
25	22, 24	eqsstrid 3960	. . . . 5 ⊢ (𝜑 → (𝐵 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
26	21, 25	unssd 4132	. . . 4 ⊢ (𝜑 → ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵))) ⊆ 𝑉)
27	17, 26	eqsstrid 3960	. . 3 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
28	16, 27	coss12d 14934	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (𝑉 ∘ 𝑉))
29	3, 28	eqsstrrd 3957	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2932   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   × cxp 5629   ∘ ccom 5635

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-co 5640

This theorem is referenced by: (None)