Theorem trrelssd 14324

Description: The composition of subclasses of a transitive relation is a subclass of that relation. (Contributed by RP, 24-Dec-2019.)

Hypotheses

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

Assertion

Ref	Expression
trrelssd	⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

Proof of Theorem trrelssd

Step	Hyp	Ref	Expression
1		trrelssd.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝑅)
2		trrelssd.t	. . 3 ⊢ (𝜑 → 𝑇 ⊆ 𝑅)
3	1, 2	coss12d 14323	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅))
4		trrelssd.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5	3, 4	sstrd 3925	1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

Syntax hints:   → wi 4   ⊆ wss 3881   ∘ ccom 5523

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-br 5031  df-opab 5093  df-co 5528

This theorem is referenced by:  trclfvlb2  14361  trrelind  40366  iunrelexpmin1  40409  iunrelexpmin2  40413