Theorem supp0cosupp0

Description: The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019)

	Ref	Expression
Assertion	supp0cosupp0	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( 𝐹 supp 𝑍 ) = ∅ → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ∅ ) )

Proof

Step	Hyp	Ref	Expression
1		suppco	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ( ◡ 𝐺 “ ( 𝐹 supp 𝑍 ) ) )
2		imaeq2	⊢ ( ( 𝐹 supp 𝑍 ) = ∅ → ( ◡ 𝐺 “ ( 𝐹 supp 𝑍 ) ) = ( ◡ 𝐺 “ ∅ ) )
3		ima0	⊢ ( ◡ 𝐺 “ ∅ ) = ∅
4	2 3	eqtrdi	⊢ ( ( 𝐹 supp 𝑍 ) = ∅ → ( ◡ 𝐺 “ ( 𝐹 supp 𝑍 ) ) = ∅ )
5	1 4	sylan9eq	⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ∧ ( 𝐹 supp 𝑍 ) = ∅ ) → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ∅ )
6	5	ex	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( 𝐹 supp 𝑍 ) = ∅ → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ∅ ) )