Theorem ustund 21835

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 5592	. . 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		xpundir 5095	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) = ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵)))
5		xpindi 5177	. . . . . . 7 ⊢ (𝐴 × (𝐴 ∩ 𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
6		inss1 3795	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
7		ustund.1	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
8	6, 7	syl5ss 3579	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
9	5, 8	syl5eqss 3612	. . . . . 6 ⊢ (𝜑 → (𝐴 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
10		xpindi 5177	. . . . . . 7 ⊢ (𝐵 × (𝐴 ∩ 𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
11		inss2 3796	. . . . . . . 8 ⊢ ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12		ustund.2	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
13	11, 12	syl5ss 3579	. . . . . . 7 ⊢ (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
14	10, 13	syl5eqss 3612	. . . . . 6 ⊢ (𝜑 → (𝐵 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
15	9, 14	unssd 3751	. . . . 5 ⊢ (𝜑 → ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵))) ⊆ 𝑉)
16	4, 15	syl5eqss 3612	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
17		coss2 5200	. . . 4 ⊢ (((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) ⊆ 𝑉 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ 𝑉))
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ 𝑉))
19		xpundi 5094	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) = (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵))
20		xpindir 5178	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
21		inss1 3795	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
22	21, 7	syl5ss 3579	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
23	20, 22	syl5eqss 3612	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐴) ⊆ 𝑉)
24		xpindir 5178	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
25		inss2 3796	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
26	25, 12	syl5ss 3579	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
27	24, 26	syl5eqss 3612	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐵) ⊆ 𝑉)
28	23, 27	unssd 3751	. . . . 5 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵)) ⊆ 𝑉)
29	19, 28	syl5eqss 3612	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ 𝑉)
30		coss1 5199	. . . 4 ⊢ (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ 𝑉 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ 𝑉) ⊆ (𝑉 ∘ 𝑉))
31	29, 30	syl 17	. . 3 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ 𝑉) ⊆ (𝑉 ∘ 𝑉))
32	18, 31	sstrd 3578	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (𝑉 ∘ 𝑉))
33	3, 32	eqsstr3d 3603	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   × cxp 5036   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-co 5047

This theorem is referenced by: (None)