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Theorem map0b 6802
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 6788 . . . 4  |-  ( f  e.  ( (/)  ^m  A
)  ->  f : A
--> (/) )
2 fdm 5359 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  A )
3 frn 5361 . . . . . . 7  |-  ( f : A --> (/)  ->  ran  f  C_  (/) )
4 ss0 3486 . . . . . . 7  |-  ( ran  f  C_  (/)  ->  ran  f  =  (/) )
53, 4syl 15 . . . . . 6  |-  ( f : A --> (/)  ->  ran  f  =  (/) )
6 dm0rn0 4894 . . . . . 6  |-  (  dom  f  =  (/)  <->  ran  f  =  (/) )
75, 6sylibr 203 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  (/) )
82, 7eqtr3d 2318 . . . 4  |-  ( f : A --> (/)  ->  A  =  (/) )
91, 8syl 15 . . 3  |-  ( f  e.  ( (/)  ^m  A
)  ->  A  =  (/) )
109necon3ai 2487 . 2  |-  ( A  =/=  (/)  ->  -.  f  e.  ( (/)  ^m  A ) )
1110eq0rdv 3490 1  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1685    =/= wne 2447    C_ wss 3153   (/)c0 3456    dom cdm 4688   ran crn 4689   -->wf 5217  (class class class)co 5820    ^m cmap 6768
This theorem is referenced by:  map0g  6803  mapdom2  7028  ply1plusgfvi  16316
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-map 6770
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