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Theorem map0g 6990
Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
map0g  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )

Proof of Theorem map0g
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 n0 3581 . . . . 5  |-  ( A  =/=  (/)  <->  E. f  f  e.  A )
2 fconst6g 5573 . . . . . . . 8  |-  ( f  e.  A  ->  ( B  X.  { f } ) : B --> A )
3 elmapg 6968 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( B  X.  { f } )  e.  ( A  ^m  B )  <->  ( B  X.  { f } ) : B --> A ) )
42, 3syl5ibr 213 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( B  X.  {
f } )  e.  ( A  ^m  B
) ) )
5 ne0i 3578 . . . . . . 7  |-  ( ( B  X.  { f } )  e.  ( A  ^m  B )  ->  ( A  ^m  B )  =/=  (/) )
64, 5syl6 31 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( A  ^m  B
)  =/=  (/) ) )
76exlimdv 1643 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. f  f  e.  A  ->  ( A  ^m  B )  =/=  (/) ) )
81, 7syl5bi 209 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  =/=  (/)  ->  ( A  ^m  B )  =/=  (/) ) )
98necon4d 2614 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  A  =  (/) ) )
10 f0 5568 . . . . . . 7  |-  (/) : (/) --> A
11 feq2 5518 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/) : B --> A  <->  (/) : (/) --> A ) )
1210, 11mpbiri 225 . . . . . 6  |-  ( B  =  (/)  ->  (/) : B --> A )
13 elmapg 6968 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (/)  e.  ( A  ^m  B )  <->  (/) : B --> A ) )
1412, 13syl5ibr 213 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  -> 
(/)  e.  ( A  ^m  B ) ) )
15 ne0i 3578 . . . . 5  |-  ( (/)  e.  ( A  ^m  B
)  ->  ( A  ^m  B )  =/=  (/) )
1614, 15syl6 31 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  ->  ( A  ^m  B
)  =/=  (/) ) )
1716necon2d 2601 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  B  =/=  (/) ) )
189, 17jcad 520 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
19 oveq1 6028 . . 3  |-  ( A  =  (/)  ->  ( A  ^m  B )  =  ( (/)  ^m  B ) )
20 map0b 6989 . . 3  |-  ( B  =/=  (/)  ->  ( (/)  ^m  B
)  =  (/) )
2119, 20sylan9eq 2440 . 2  |-  ( ( A  =  (/)  /\  B  =/=  (/) )  ->  ( A  ^m  B )  =  (/) )
2218, 21impbid1 195 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2551   (/)c0 3572   {csn 3758    X. cxp 4817   -->wf 5391  (class class class)co 6021    ^m cmap 6955
This theorem is referenced by:  map0  6991  mapdom2  7215
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-map 6957
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