Theorem trrelssd 15094

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 15093	. 2 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ (𝑅 ∘ 𝑅))
4		trrelssd.r	. 2 ⊢ (𝜑 → (𝑅 ∘ 𝑅) ⊆ 𝑅)
5	3, 4	sstrd 3940	1 ⊢ (𝜑 → (𝑆 ∘ 𝑇) ⊆ 𝑅)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3898   ∘ ccom 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3915  df-br 5103  df-opab 5167  df-co 5656

This theorem is used by:  trclfvlb2  15131  trrelind  44609  iunrelexpmin1  44652  iunrelexpmin2  44656