MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  map0b Structured version   Visualization version   Unicode version

Theorem map0b 7528
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.
StepHypRef Expression
1 elmapi 7511 . . . 4  |-  ( f  e.  ( (/)  ^m  A
)  ->  f : A
--> (/) )
2 fdm 5745 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  A )
3 frn 5747 . . . . . . 7  |-  ( f : A --> (/)  ->  ran  f  C_  (/) )
4 ss0 3768 . . . . . . 7  |-  ( ran  f  C_  (/)  ->  ran  f  =  (/) )
53, 4syl 17 . . . . . 6  |-  ( f : A --> (/)  ->  ran  f  =  (/) )
6 dm0rn0 5057 . . . . . 6  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
75, 6sylibr 217 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  (/) )
82, 7eqtr3d 2507 . . . 4  |-  ( f : A --> (/)  ->  A  =  (/) )
91, 8syl 17 . . 3  |-  ( f  e.  ( (/)  ^m  A
)  ->  A  =  (/) )
109necon3ai 2668 . 2  |-  ( A  =/=  (/)  ->  -.  f  e.  ( (/)  ^m  A ) )
1110eq0rdv 3773 1  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    =/= wne 2641    C_ wss 3390   (/)c0 3722   dom cdm 4839   ran crn 4840   -->wf 5585  (class class class)co 6308    ^m cmap 7490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-map 7492
This theorem is referenced by:  map0g  7529  mapdom2  7761  ply1plusgfvi  18912
  Copyright terms: Public domain W3C validator