Theorem ustund 22025

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 5675	. . 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 5172	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) = ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵)))
5		xpindi 5255	. . . . . . 7 ⊢ (𝐴 × (𝐴 ∩ 𝐵)) = ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵))
6		inss1 3833	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ (𝐴 × 𝐴)
7		ustund.1	. . . . . . . 8 ⊢ (𝜑 → (𝐴 × 𝐴) ⊆ 𝑉)
8	6, 7	syl5ss 3614	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐴 × 𝐵)) ⊆ 𝑉)
9	5, 8	syl5eqss 3649	. . . . . 6 ⊢ (𝜑 → (𝐴 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
10		xpindi 5255	. . . . . . 7 ⊢ (𝐵 × (𝐴 ∩ 𝐵)) = ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵))
11		inss2 3834	. . . . . . . 8 ⊢ ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
12		ustund.2	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) ⊆ 𝑉)
13	11, 12	syl5ss 3614	. . . . . . 7 ⊢ (𝜑 → ((𝐵 × 𝐴) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
14	10, 13	syl5eqss 3649	. . . . . 6 ⊢ (𝜑 → (𝐵 × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
15	9, 14	unssd 3789	. . . . 5 ⊢ (𝜑 → ((𝐴 × (𝐴 ∩ 𝐵)) ∪ (𝐵 × (𝐴 ∩ 𝐵))) ⊆ 𝑉)
16	4, 15	syl5eqss 3649	. . . 4 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵)) ⊆ 𝑉)
17		coss2 5278	. . . 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 5171	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) = (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵))
20		xpindir 5256	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) × 𝐴) = ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴))
21		inss1 3833	. . . . . . . 8 ⊢ ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ (𝐴 × 𝐴)
22	21, 7	syl5ss 3614	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐴) ∩ (𝐵 × 𝐴)) ⊆ 𝑉)
23	20, 22	syl5eqss 3649	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐴) ⊆ 𝑉)
24		xpindir 5256	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) × 𝐵) = ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵))
25		inss2 3834	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ (𝐵 × 𝐵)
26	25, 12	syl5ss 3614	. . . . . . 7 ⊢ (𝜑 → ((𝐴 × 𝐵) ∩ (𝐵 × 𝐵)) ⊆ 𝑉)
27	24, 26	syl5eqss 3649	. . . . . 6 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × 𝐵) ⊆ 𝑉)
28	23, 27	unssd 3789	. . . . 5 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × 𝐴) ∪ ((𝐴 ∩ 𝐵) × 𝐵)) ⊆ 𝑉)
29	19, 28	syl5eqss 3649	. . . 4 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ 𝑉)
30		coss1 5277	. . . 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 3613	. 2 ⊢ (𝜑 → (((𝐴 ∩ 𝐵) × (𝐴 ∪ 𝐵)) ∘ ((𝐴 ∪ 𝐵) × (𝐴 ∩ 𝐵))) ⊆ (𝑉 ∘ 𝑉))
33	3, 32	eqsstr3d 3640	1 ⊢ (𝜑 → ((𝐴 ∪ 𝐵) × (𝐴 ∪ 𝐵)) ⊆ (𝑉 ∘ 𝑉))

Syntax hints:   → wi 4   = wceq 1483   ≠ wne 2794   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915   × cxp 5112   ∘ ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by: (None)