Theorem trrelssd 13560

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 13559	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅))
4		trrelssd.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5	3, 4	sstrd 3578	1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

Syntax hints:   → wi 4   ⊆ wss 3540   ∘ ccom 5042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-co 5047

This theorem is referenced by:  trclfvlb2  13599  trrelind  36976  iunrelexpmin1  37019  iunrelexpmin2  37023