Theorem trrelind 43568

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 4252	. . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑅
3	2	a1i 11	. . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑅)
4	1, 3, 3	trrelssd 15018	. . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑅)
5		trrelind.s	. . . 4 ⊢ (𝜑 → (𝑆 ∘ 𝑆) ⊆ 𝑆)
6		inss2 4253	. . . . 5 ⊢ (𝑅 ∩ 𝑆) ⊆ 𝑆
7	6	a1i 11	. . . 4 ⊢ (𝜑 → (𝑅 ∩ 𝑆) ⊆ 𝑆)
8	5, 7, 7	trrelssd 15018	. . 3 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ 𝑆)
9	4, 8	ssind 4256	. 2 ⊢ (𝜑 → ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)) ⊆ (𝑅 ∩ 𝑆))
10		trrelind.t	. . 3 ⊢ (𝜑 → 𝑇 = (𝑅 ∩ 𝑆))
11	10, 10	coeq12d 5888	. 2 ⊢ (𝜑 → (𝑇 ∘ 𝑇) = ((𝑅 ∩ 𝑆) ∘ (𝑅 ∩ 𝑆)))
12	9, 11, 10	3sstr4d 4050	1 ⊢ (𝜑 → (𝑇 ∘ 𝑇) ⊆ 𝑇)

Syntax hints:   → wi 4   = wceq 1537   ∩ cin 3969   ⊆ wss 3970   ∘ ccom 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-in 3977  df-ss 3987  df-br 5170  df-opab 5232  df-co 5708

This theorem is referenced by:  xpintrreld  43569