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Metamath Proof Explorer

Theorem mpompt

Description: Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013) (Revised by Mario Carneiro, 29-Dec-2014)

		Ref	Expression
	Hypothesis	mpompt.1	$⊢ z = ⟨x, y⟩ \to C = D$
	Assertion	mpompt	$⊢ (z \in (A \times B) ⟼ C) = (x \in A, y \in B ⟼ D)$

Proof

Step	Hyp	Ref	Expression
1		mpompt.1	$⊢ z = ⟨x, y⟩ \to C = D$
2		iunxpconst	$⊢ ⋃_{x \in A} (\{x\} \times B) = A \times B$
3	2	mpteq1i	$⊢ (z \in ⋃_{x \in A} (\{x\} \times B) ⟼ C) = (z \in (A \times B) ⟼ C)$
4	1	mpomptx	$⊢ (z \in ⋃_{x \in A} (\{x\} \times B) ⟼ C) = (x \in A, y \in B ⟼ D)$
5	3 4	eqtr3i	$⊢ (z \in (A \times B) ⟼ C) = (x \in A, y \in B ⟼ D)$