Theorem map0g 6713

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		fconst6g 5433	. . . . . . . 8 |- ( f e. A -> ( B X. { f } ) : B --> A )
2		elmapg 6686	. . . . . . . 8 |- ( ( A e. V /\ B e. W ) -> ( ( B X. { f } ) e. ( A ^m B ) <-> ( B X. { f } ) : B --> A ) )
3	1, 2	imbitrrid 156	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( B X. { f } ) e. ( A ^m B ) ) )
4		ne0i 3444	. . . . . . 7 |- ( ( B X. { f } ) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
5	3, 4	syl6 33	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( f e. A -> ( A ^m B ) =/= (/) ) )
6	5	exlimdv 1830	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
7	6	necon2bd 2418	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> -. E. f f e. A ) )
8		notm0 3458	. . . 4 |- ( -. E. f f e. A <-> A = (/) )
9	7, 8	imbitrdi 161	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> A = (/) ) )
10		f0 5425	. . . . . . 7 |- (/) : (/) --> A
11		feq2 5368	. . . . . . 7 |- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10, 11	mpbiri 168	. . . . . 6 |- ( B = (/) -> (/) : B --> A )
13		elmapg 6686	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( (/) e. ( A ^m B ) <-> (/) : B --> A ) )
14	12, 13	imbitrrid 156	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( B = (/) -> (/) e. ( A ^m B ) ) )
15		ne0i 3444	. . . . 5 |- ( (/) e. ( A ^m B ) -> ( A ^m B ) =/= (/) )
16	14, 15	syl6 33	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( B = (/) -> ( A ^m B ) =/= (/) ) )
17	16	necon2d 2419	. . 3 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> B =/= (/) ) )
18	9, 17	jcad 307	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> ( A = (/) /\ B =/= (/) ) ) )
19		oveq1 5902	. . 3 |- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b 6712	. . 3 |- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19, 20	sylan9eq 2242	. 2 |- ( ( A = (/) /\ B =/= (/) ) -> ( A ^m B ) = (/) )
22	18, 21	impbid1 142	1 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) <-> ( A = (/) /\ B =/= (/) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   E.wex 1503   e. wcel 2160   =/= wne 2360   (/)c0 3437   {csn 3607   X. cxp 4642   -->wf 5231  (class class class)co 5895   ^m cmap 6673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-map 6675

This theorem is referenced by:  map0  6714