Theorem coexd

Description: The composition of two sets is a set. (Contributed by SN, 7-Feb-2025)

	Ref	Expression
Hypotheses	coexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	coexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
Assertion	coexd	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐵 ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		coexd.1	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		coexd.2	⊢ ( 𝜑 → 𝐵 ∈ 𝑊 )
3		coexg	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∘ 𝐵 ) ∈ V )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐵 ) ∈ V )