Theorem trrelssd 14924

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 14923	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅))
4		trrelssd.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5	3, 4	sstrd 3927	1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

Syntax hints:   → wi 4   ⊆ wss 3885   ∘ ccom 5624

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ss 3902  df-br 5075  df-opab 5137  df-co 5629

This theorem is referenced by:  trclfvlb2  14961  trrelind  44080  iunrelexpmin1  44123  iunrelexpmin2  44127