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Metamath Proof Explorer

Theorem mapsnen

Description: Set exponentiation to a singleton exponent is equinumerous to its base. Exercise 4.43 of Mendelson p. 255. (Contributed by NM, 17-Dec-2003) (Revised by Mario Carneiro, 15-Nov-2014) (Proof shortened by AV, 17-Jul-2022)

	Ref	Expression
Hypotheses	mapsnen.1	⊢ 𝐴 ∈ V
	mapsnen.2	⊢ 𝐵 ∈ V
Assertion	mapsnen	⊢ ( 𝐴 ↑_m { 𝐵 } ) ≈ 𝐴

Proof

Step	Hyp	Ref	Expression
1		mapsnen.1	⊢ 𝐴 ∈ V
2		mapsnen.2	⊢ 𝐵 ∈ V
3		id	⊢ ( 𝐴 ∈ V → 𝐴 ∈ V )
4	2	a1i	⊢ ( 𝐴 ∈ V → 𝐵 ∈ V )
5	3 4	mapsnend	⊢ ( 𝐴 ∈ V → ( 𝐴 ↑_m { 𝐵 } ) ≈ 𝐴 )
6	1 5	ax-mp	⊢ ( 𝐴 ↑_m { 𝐵 } ) ≈ 𝐴