Theorem trrelind 43704

Description: The intersection of transitive relations is a transitive relation. (Contributed by RP, 24-Dec-2019.)

Hypotheses

Ref	Expression
trrelind.r	⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
trrelind.s	⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
trrelind.t	⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆))

Assertion

Ref	Expression
trrelind	⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇)

Proof of Theorem trrelind

Step	Hyp	Ref	Expression
1		trrelind.r	. . . 4 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
2		inss1 4187	. . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑅
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑅)
4	1, 3, 3	trrelssd 14880	. . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑅)
5		trrelind.s	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
6		inss2 4188	. . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆
7	6	a1i 11	. . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑆)
8	5, 7, 7	trrelssd 14880	. . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑆)
9	4, 8	ssind 4191	. 2 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))
10		trrelind.t	. . 3 ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆))
11	10, 10	coeq12d 5804	. 2 ⊢ (𝜑 → (𝑇 ∘ 𝑇) = ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)))
12	9, 11, 10	3sstr4d 3990	1 ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇)

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3901   ⊆ wss 3902   ∘ ccom 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3909  df-ss 3919  df-br 5092  df-opab 5154  df-co 5625

This theorem is referenced by:  xpintrreld  43705