Theorem map0g 6798

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 5496	. . . . . . . 8 |- ( f e. A -> ( B X. { f } ) : B --> A )
2		elmapg 6771	. . . . . . . 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 3475	. . . . . . 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 1843	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
7	6	necon2bd 2436	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> -. E. f f e. A ) )
8		notm0 3489	. . . 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 5488	. . . . . . 7 |- (/) : (/) --> A
11		feq2 5429	. . . . . . 7 |- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10, 11	mpbiri 168	. . . . . 6 |- ( B = (/) -> (/) : B --> A )
13		elmapg 6771	. . . . . 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 3475	. . . . 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 2437	. . 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 5974	. . 3 |- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b 6797	. . 3 |- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19, 20	sylan9eq 2260	. 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 1373   E.wex 1516   e. wcel 2178   =/= wne 2378   (/)c0 3468   {csn 3643   X. cxp 4691   -->wf 5286  (class class class)co 5967   ^m cmap 6758

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-map 6760

This theorem is referenced by:  map0  6799