Theorem map0g 6835

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 5524	. . . . . . . 8 |- ( f e. A -> ( B X. { f } ) : B --> A )
2		elmapg 6808	. . . . . . . 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 3498	. . . . . . 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 1865	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
7	6	necon2bd 2458	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> -. E. f f e. A ) )
8		notm0 3512	. . . 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 5516	. . . . . . 7 |- (/) : (/) --> A
11		feq2 5457	. . . . . . 7 |- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10, 11	mpbiri 168	. . . . . 6 |- ( B = (/) -> (/) : B --> A )
13		elmapg 6808	. . . . . 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 3498	. . . . 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 2459	. . 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 6008	. . 3 |- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b 6834	. . 3 |- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19, 20	sylan9eq 2282	. 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 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   (/)c0 3491   {csn 3666   X. cxp 4717   -->wf 5314  (class class class)co 6001   ^m cmap 6795

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-map 6797

This theorem is referenced by:  map0  6836