Theorem trrelssd 14944

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 14943	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅))
4		trrelssd.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5	3, 4	sstrd 3988	1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

Syntax hints:   → wi 4   ⊆ wss 3944   ∘ ccom 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3471  df-in 3951  df-ss 3961  df-br 5143  df-opab 5205  df-co 5681

This theorem is referenced by:  trclfvlb2  14981  trrelind  43018  iunrelexpmin1  43061  iunrelexpmin2  43065