Theorem mapsn

Description: The value of set exponentiation with a singleton exponent. Theorem 98 of Suppes p. 89. (Contributed by NM, 10-Dec-2003) (Proof shortened by AV, 17-Jul-2022)

	Ref	Expression
Hypotheses	map0.1	⊢ 𝐴 ∈ V
	map0.2	⊢ 𝐵 ∈ V
Assertion	mapsn	⊢ ( 𝐴 ↑m { 𝐵 } ) = { 𝑓 ∣ ∃ 𝑦 ∈ 𝐴 𝑓 = { ⟨ 𝐵 , 𝑦 ⟩ } }

Proof

Step	Hyp	Ref	Expression
1		map0.1	⊢ 𝐴 ∈ V
2		map0.2	⊢ 𝐵 ∈ V
3		id	⊢ ( 𝐴 ∈ V → 𝐴 ∈ V )
4	2	a1i	⊢ ( 𝐴 ∈ V → 𝐵 ∈ V )
5	3 4	mapsnd	⊢ ( 𝐴 ∈ V → ( 𝐴 ↑m { 𝐵 } ) = { 𝑓 ∣ ∃ 𝑦 ∈ 𝐴 𝑓 = { ⟨ 𝐵 , 𝑦 ⟩ } } )
6	1 5	ax-mp	⊢ ( 𝐴 ↑m { 𝐵 } ) = { 𝑓 ∣ ∃ 𝑦 ∈ 𝐴 𝑓 = { ⟨ 𝐵 , 𝑦 ⟩ } }