Theorem map0b 6743

Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
map0b	|- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Proof of Theorem map0b

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapi 6726	. . . 4 |- ( f e. ( (/) ^m A ) -> f : A --> (/) )
2		fdm 5410	. . . . 5 |- ( f : A --> (/) -> dom f = A )
3		frn 5413	. . . . . . 7 |- ( f : A --> (/) -> ran f C_ (/) )
4		ss0 3488	. . . . . . 7 |- ( ran f C_ (/) -> ran f = (/) )
5	3, 4	syl 14	. . . . . 6 |- ( f : A --> (/) -> ran f = (/) )
6		dm0rn0 4880	. . . . . 6 |- ( dom f = (/) <-> ran f = (/) )
7	5, 6	sylibr 134	. . . . 5 |- ( f : A --> (/) -> dom f = (/) )
8	2, 7	eqtr3d 2228	. . . 4 |- ( f : A --> (/) -> A = (/) )
9	1, 8	syl 14	. . 3 |- ( f e. ( (/) ^m A ) -> A = (/) )
10	9	necon3ai 2413	. 2 |- ( A =/= (/) -> -. f e. ( (/) ^m A ) )
11	10	eq0rdv 3492	1 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   =/= wne 2364   C_ wss 3154   (/)c0 3447   dom cdm 4660   ran crn 4661   -->wf 5251  (class class class)co 5919   ^m cmap 6704

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-map 6706

This theorem is referenced by:  map0g  6744