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Metamath Proof Explorer

Theorem mapval

Description: The value of set exponentiation (inference version). ( A ^m B ) is the set of all functions that map from B to A . Definition 10.24 of Kunen p. 24. (Contributed by NM, 8-Dec-2003)

	Ref	Expression
Hypotheses	mapval.1	⊢ 𝐴 ∈ V
	mapval.2	⊢ 𝐵 ∈ V
Assertion	mapval	⊢ ( 𝐴 ↑_m 𝐵 ) = { 𝑓 ∣ 𝑓 : 𝐵 ⟶ 𝐴 }

Proof

Step	Hyp	Ref	Expression
1		mapval.1	⊢ 𝐴 ∈ V
2		mapval.2	⊢ 𝐵 ∈ V
3		mapvalg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ↑_m 𝐵 ) = { 𝑓 ∣ 𝑓 : 𝐵 ⟶ 𝐴 } )
4	1 2 3	mp2an	⊢ ( 𝐴 ↑_m 𝐵 ) = { 𝑓 ∣ 𝑓 : 𝐵 ⟶ 𝐴 }