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Theorem map0b 7045
Description: Set exponentiation with an empty base is the empty set, provided the exponent is non-empty. 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.
StepHypRef Expression
1 elmapi 7031 . . . 4  |-  ( f  e.  ( (/)  ^m  A
)  ->  f : A
--> (/) )
2 fdm 5588 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  A )
3 frn 5590 . . . . . . 7  |-  ( f : A --> (/)  ->  ran  f  C_  (/) )
4 ss0 3651 . . . . . . 7  |-  ( ran  f  C_  (/)  ->  ran  f  =  (/) )
53, 4syl 16 . . . . . 6  |-  ( f : A --> (/)  ->  ran  f  =  (/) )
6 dm0rn0 5079 . . . . . 6  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
75, 6sylibr 204 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  (/) )
82, 7eqtr3d 2470 . . . 4  |-  ( f : A --> (/)  ->  A  =  (/) )
91, 8syl 16 . . 3  |-  ( f  e.  ( (/)  ^m  A
)  ->  A  =  (/) )
109necon3ai 2639 . 2  |-  ( A  =/=  (/)  ->  -.  f  e.  ( (/)  ^m  A ) )
1110eq0rdv 3655 1  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3313   (/)c0 3621   dom cdm 4871   ran crn 4872   -->wf 5443  (class class class)co 6074    ^m cmap 7011
This theorem is referenced by:  map0g  7046  mapdom2  7271  ply1plusgfvi  16629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-map 7013
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