Theorem coindirss

Description: Precomposition is subdistributive over intersection. Compare with coundir . See coindir for equality when precomposing with a single-valued class. (Contributed by BJ, 8-Nov-2026)

	Ref	Expression
Assertion	coindirss	⊢ ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( ( 𝐴 ∘ 𝐶 ) ∩ ( 𝐵 ∘ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		inss1	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴
2		coss1	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 → ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( 𝐴 ∘ 𝐶 ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( 𝐴 ∘ 𝐶 )
4		inss2	⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵
5		coss1	⊢ ( ( 𝐴 ∩ 𝐵 ) ⊆ 𝐵 → ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 ) )
6	4 5	ax-mp	⊢ ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( 𝐵 ∘ 𝐶 )
7	3 6	ssini	⊢ ( ( 𝐴 ∩ 𝐵 ) ∘ 𝐶 ) ⊆ ( ( 𝐴 ∘ 𝐶 ) ∩ ( 𝐵 ∘ 𝐶 ) )