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Metamath Proof Explorer

Theorem fmptsnxp

Description: Maps-to notation and Cartesian product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Assertion fmptsnxp ⊢ A ∈ V ∧ B ∈ W → x ∈ A ⟼ B = A × B

Proof

Step Hyp Ref Expression
1 xpsng ⊢ A ∈ V ∧ B ∈ W → A × B = A B
2 fmptsn ⊢ A ∈ V ∧ B ∈ W → A B = x ∈ A ⟼ B
3 1 2 eqtr2d ⊢ A ∈ V ∧ B ∈ W → x ∈ A ⟼ B = A × B