Theorem map0g 6690

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 5416	. . . . . . . 8 |- ( f e. A -> ( B X. { f } ) : B --> A )
2		elmapg 6663	. . . . . . . 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 3431	. . . . . . 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 1819	. . . . 5 |- ( ( A e. V /\ B e. W ) -> ( E. f f e. A -> ( A ^m B ) =/= (/) ) )
7	6	necon2bd 2405	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( ( A ^m B ) = (/) -> -. E. f f e. A ) )
8		notm0 3445	. . . 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 5408	. . . . . . 7 |- (/) : (/) --> A
11		feq2 5351	. . . . . . 7 |- ( B = (/) -> ( (/) : B --> A <-> (/) : (/) --> A ) )
12	10, 11	mpbiri 168	. . . . . 6 |- ( B = (/) -> (/) : B --> A )
13		elmapg 6663	. . . . . 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 3431	. . . . 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 2406	. . 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 5884	. . 3 |- ( A = (/) -> ( A ^m B ) = ( (/) ^m B ) )
20		map0b 6689	. . 3 |- ( B =/= (/) -> ( (/) ^m B ) = (/) )
21	19, 20	sylan9eq 2230	. 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 1353   E.wex 1492   e. wcel 2148   =/= wne 2347   (/)c0 3424   {csn 3594   X. cxp 4626   -->wf 5214  (class class class)co 5877   ^m cmap 6650

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-map 6652

This theorem is referenced by:  map0  6691