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Theorem map0b 7015
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 7001 . . . 4  |-  ( f  e.  ( (/)  ^m  A
)  ->  f : A
--> (/) )
2 fdm 5558 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  A )
3 frn 5560 . . . . . . 7  |-  ( f : A --> (/)  ->  ran  f  C_  (/) )
4 ss0 3622 . . . . . . 7  |-  ( ran  f  C_  (/)  ->  ran  f  =  (/) )
53, 4syl 16 . . . . . 6  |-  ( f : A --> (/)  ->  ran  f  =  (/) )
6 dm0rn0 5049 . . . . . 6  |-  ( dom  f  =  (/)  <->  ran  f  =  (/) )
75, 6sylibr 204 . . . . 5  |-  ( f : A --> (/)  ->  dom  f  =  (/) )
82, 7eqtr3d 2442 . . . 4  |-  ( f : A --> (/)  ->  A  =  (/) )
91, 8syl 16 . . 3  |-  ( f  e.  ( (/)  ^m  A
)  ->  A  =  (/) )
109necon3ai 2611 . 2  |-  ( A  =/=  (/)  ->  -.  f  e.  ( (/)  ^m  A ) )
1110eq0rdv 3626 1  |-  ( A  =/=  (/)  ->  ( (/)  ^m  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    =/= wne 2571    C_ wss 3284   (/)c0 3592   dom cdm 4841   ran crn 4842   -->wf 5413  (class class class)co 6044    ^m cmap 6981
This theorem is referenced by:  map0g  7016  mapdom2  7241  ply1plusgfvi  16595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-map 6983
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