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Metamath Proof Explorer

Theorem mapsspw

Description: Set exponentiation is a subset of the power set of the Cartesian product of its arguments. (Contributed by NM, 8-Dec-2006) (Revised by Mario Carneiro, 26-Apr-2015)

		Ref	Expression
	Assertion	mapsspw	⊢ ( 𝐴 ↑_m 𝐵 ) ⊆ 𝒫 ( 𝐵 × 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mapsspm	⊢ ( 𝐴 ↑_m 𝐵 ) ⊆ ( 𝐴 ↑_pm 𝐵 )
2		pmsspw	⊢ ( 𝐴 ↑_pm 𝐵 ) ⊆ 𝒫 ( 𝐵 × 𝐴 )
3	1 2	sstri	⊢ ( 𝐴 ↑_m 𝐵 ) ⊆ 𝒫 ( 𝐵 × 𝐴 )