Theorem map0g 7897

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.)

Assertion

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

Proof of Theorem map0g

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . . . 5 |- ( A =/= (/) <-> E. f f e. A )
2		fconst6g 6094	. . . . . . . 8 |- ( f e. A -> ( B X. { f } ) : B --> A )
3		elmapg 7870	. . . . . . . 8 |- ( ( A e. V /\ B e. W ) -> ( ( B X. { f } ) e. ( A ^m B ) <-> ( B X. { f } ) : B --> A ) )
4	2, 3	syl5ibr 236	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( B X. { f } ) e. ( A ^m B ) ) )
5		ne0i 3921	. . . . . . 7 |- ( ( B X. { f } ) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
6	4, 5	syl6 35	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( A ^m B ) =/= (/) ) )
7	6	exlimdv 1861	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
8	1, 7	syl5bi 232	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( A =/= (/) -> ( A ^m B ) =/= (/) ) )
9	8	necon4d 2818	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> A = (/) ) )
10		f0 6086	. . . . . . 7 |- (/) : (/) --> A
11		feq2 6027	. . . . . . 7 |- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10, 11	mpbiri 248	. . . . . 6 |- ( B = (/) -> (/) : B --> A )
13		elmapg 7870	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( (/) e. ( A ^m B ) <-> (/) : B --> A ) )
14	12, 13	syl5ibr 236	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( B = (/) -> (/) e. ( A ^m B ) ) )
15		ne0i 3921	. . . . 5 |- ( (/) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
16	14, 15	syl6 35	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( B = (/) -> ( A ^m B ) =/= (/) ) )
17	16	necon2d 2817	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> B =/= (/) ) )
18	9, 17	jcad 555	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> ( A = (/) /\ B =/= (/) ) ) )
19		oveq1 6657	. . 3 |- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b 7896	. . 3 |- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19, 20	sylan9eq 2676	. 2 |- ( ( A = (/) /\ B =/= (/) ) -> ( A ^m B ) = (/) )
22	18, 21	impbid1 215	1 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   {csn 4177   X. cxp 5112   -->wf 5884  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  map0  7898  mapdom2  8131