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Theorem map0b 6806
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 6792 . . . 4  |-  ( f  e.  ( (/)  ^m  A
)  ->  f : A
--> (/) )
2 fdm 5393 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  A )
3 frn 5395 . . . . . . 7  |-  ( f : A --> (/)  ->  ran  f  C_  (/) )
4 ss0 3485 . . . . . . 7  |-  ( ran  f  C_  (/)  ->  ran  f  =  (/) )
53, 4syl 15 . . . . . 6  |-  ( f : A --> (/)  ->  ran  f  =  (/) )
6 dm0rn0 4895 . . . . . 6  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
75, 6sylibr 203 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  (/) )
82, 7eqtr3d 2317 . . . 4  |-  ( f : A --> (/)  ->  A  =  (/) )
91, 8syl 15 . . 3  |-  ( f  e.  ( (/)  ^m  A
)  ->  A  =  (/) )
109necon3ai 2486 . 2  |-  ( A  =/=  (/)  ->  -.  f  e.  ( (/)  ^m  A ) )
1110eq0rdv 3489 1  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   dom cdm 4689   ran crn 4690   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  map0g  6807  mapdom2  7032  ply1plusgfvi  16320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774
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