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Metamath Proof Explorer

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 )