Theorem coss1

Description: Subclass theorem for composition. (Contributed by FL, 30-Dec-2010)

	Ref	Expression
Assertion	coss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssbr	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑦 𝐴 𝑧 → 𝑦 𝐵 𝑧 ) )
2	1	anim2d	⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐴 𝑧 ) → ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐵 𝑧 ) ) )
3	2	eximdv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐴 𝑧 ) → ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐵 𝑧 ) ) )
4	3	ssopab2dv	⊢ ( 𝐴 ⊆ 𝐵 → { ⟨ 𝑥 , 𝑧 ⟩ ∣ ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐴 𝑧 ) } ⊆ { ⟨ 𝑥 , 𝑧 ⟩ ∣ ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐵 𝑧 ) } )
5		df-co	⊢ ( 𝐴 ∘ 𝐶 ) = { ⟨ 𝑥 , 𝑧 ⟩ ∣ ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐴 𝑧 ) }
6		df-co	⊢ ( 𝐵 ∘ 𝐶 ) = { ⟨ 𝑥 , 𝑧 ⟩ ∣ ∃ 𝑦 ( 𝑥 𝐶 𝑦 ∧ 𝑦 𝐵 𝑧 ) }
7	4 5 6	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )