Theorem map0g

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of Suppes p. 89. (Contributed by Mario Carneiro, 30-Apr-2015)

	Ref	Expression
Assertion	map0g	|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )

Proof

Step	Hyp	Ref	Expression
1		n0	|- ( A =/= (/) <-> E. f f e. A )
2		fconst6g	|- ( f e. A -> ( B X. { f } ) : B --> A )
3		elmapg	|- ( ( A e. V /\ B e. W ) -> ( ( B X. { f } ) e. ( A ^m B ) <-> ( B X. { f } ) : B --> A ) )
4	2 3	syl5ibr	|- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( B X. { f } ) e. ( A ^m B ) ) )
5		ne0i	|- ( ( B X. { f } ) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
6	4 5	syl6	|- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( A ^m B ) =/= (/) ) )
7	6	exlimdv	|- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
8	1 7	syl5bi	|- ( ( A e. V /\ B e. W ) -> ( A =/= (/) -> ( A ^m B ) =/= (/) ) )
9	8	necon4d	|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> A = (/) ) )
10		f0	|- (/) : (/) --> A
11		feq2	|- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10 11	mpbiri	|- ( B = (/) -> (/) : B --> A )
13		elmapg	|- ( ( A e. V /\ B e. W ) -> ( (/) e. ( A ^m B ) <-> (/) : B --> A ) )
14	12 13	syl5ibr	|- ( ( A e. V /\ B e. W ) -> ( B = (/) -> (/) e. ( A ^m B ) ) )
15		ne0i	|- ( (/) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
16	14 15	syl6	|- ( ( A e. V /\ B e. W ) -> ( B = (/) -> ( A ^m B ) =/= (/) ) )
17	16	necon2d	|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> B =/= (/) ) )
18	9 17	jcad	|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> ( A = (/) /\ B =/= (/) ) ) )
19		oveq1	|- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b	|- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19 20	sylan9eq	|- ( ( A = (/) /\ B =/= (/) ) -> ( A ^m B ) = (/) )
22	18 21	impbid1	|- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )