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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 e. _V
	mapsnen.2	\|- B e. _V
Assertion	mapsnen	\|- ( A ^m { B } ) ~~ A

Proof

Step	Hyp	Ref	Expression
1		mapsnen.1	\|- A e. _V
2		mapsnen.2	\|- B e. _V
3		id	\|- ( A e. _V -> A e. _V )
4	2	a1i	\|- ( A e. _V -> B e. _V )
5	3 4	mapsnend	\|- ( A e. _V -> ( A ^m { B } ) ~~ A )
6	1 5	ax-mp	\|- ( A ^m { B } ) ~~ A