Theorem map0b 6684

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 6667	. . . 4 |- ( f e. ( (/) ^m A ) -> f : A --> (/) )
2		fdm 5370	. . . . 5 |- ( f : A --> (/) -> dom f = A )
3		frn 5373	. . . . . . 7 |- ( f : A --> (/) -> ran f C_ (/) )
4		ss0 3463	. . . . . . 7 |- ( ran f C_ (/) -> ran f = (/) )
5	3, 4	syl 14	. . . . . 6 |- ( f : A --> (/) -> ran f = (/) )
6		dm0rn0 4843	. . . . . 6 |- ( dom f = (/) <-> ran f = (/) )
7	5, 6	sylibr 134	. . . . 5 |- ( f : A --> (/) -> dom f = (/) )
8	2, 7	eqtr3d 2212	. . . 4 |- ( f : A --> (/) -> A = (/) )
9	1, 8	syl 14	. . 3 |- ( f e. ( (/) ^m A ) -> A = (/) )
10	9	necon3ai 2396	. 2 |- ( A =/= (/) -> -. f e. ( (/) ^m A ) )
11	10	eq0rdv 3467	1 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   =/= wne 2347   C_ wss 3129   (/)c0 3422   dom cdm 4625   ran crn 4626   -->wf 5211  (class class class)co 5872   ^m cmap 6645

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 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535

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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-fv 5223  df-ov 5875  df-oprab 5876  df-mpo 5877  df-map 6647

This theorem is referenced by:  map0g  6685