Theorem map0b 6712

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 6695	. . . 4 |- ( f e. ( (/) ^m A ) -> f : A --> (/) )
2		fdm 5390	. . . . 5 |- ( f : A --> (/) -> dom f = A )
3		frn 5393	. . . . . . 7 |- ( f : A --> (/) -> ran f C_ (/) )
4		ss0 3478	. . . . . . 7 |- ( ran f C_ (/) -> ran f = (/) )
5	3, 4	syl 14	. . . . . 6 |- ( f : A --> (/) -> ran f = (/) )
6		dm0rn0 4862	. . . . . 6 |- ( dom f = (/) <-> ran f = (/) )
7	5, 6	sylibr 134	. . . . 5 |- ( f : A --> (/) -> dom f = (/) )
8	2, 7	eqtr3d 2224	. . . 4 |- ( f : A --> (/) -> A = (/) )
9	1, 8	syl 14	. . 3 |- ( f e. ( (/) ^m A ) -> A = (/) )
10	9	necon3ai 2409	. 2 |- ( A =/= (/) -> -. f e. ( (/) ^m A ) )
11	10	eq0rdv 3482	1 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   =/= wne 2360   C_ wss 3144   (/)c0 3437   dom cdm 4644   ran crn 4645   -->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-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-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:  map0g  6713