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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