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Theorem map0g 6807
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 3464 . . . . 5  |-  ( A  =/=  (/)  <->  E. f  f  e.  A )
2 fconst6g 5430 . . . . . . . 8  |-  ( f  e.  A  ->  ( B  X.  { f } ) : B --> A )
3 elmapg 6785 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( B  X.  { f } )  e.  ( A  ^m  B )  <->  ( B  X.  { f } ) : B --> A ) )
42, 3syl5ibr 212 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( B  X.  {
f } )  e.  ( A  ^m  B
) ) )
5 ne0i 3461 . . . . . . 7  |-  ( ( B  X.  { f } )  e.  ( A  ^m  B )  ->  ( A  ^m  B )  =/=  (/) )
64, 5syl6 29 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( A  ^m  B
)  =/=  (/) ) )
76exlimdv 1664 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. f  f  e.  A  ->  ( A  ^m  B )  =/=  (/) ) )
81, 7syl5bi 208 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  =/=  (/)  ->  ( A  ^m  B )  =/=  (/) ) )
98necon4d 2509 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  A  =  (/) ) )
10 f0 5425 . . . . . . 7  |-  (/) : (/) --> A
11 feq2 5376 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/) : B --> A  <->  (/) : (/) --> A ) )
1210, 11mpbiri 224 . . . . . 6  |-  ( B  =  (/)  ->  (/) : B --> A )
13 elmapg 6785 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (/)  e.  ( A  ^m  B )  <->  (/) : B --> A ) )
1412, 13syl5ibr 212 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  -> 
(/)  e.  ( A  ^m  B ) ) )
15 ne0i 3461 . . . . 5  |-  ( (/)  e.  ( A  ^m  B
)  ->  ( A  ^m  B )  =/=  (/) )
1614, 15syl6 29 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  ->  ( A  ^m  B
)  =/=  (/) ) )
1716necon2d 2496 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  B  =/=  (/) ) )
189, 17jcad 519 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
19 oveq1 5865 . . 3  |-  ( A  =  (/)  ->  ( A  ^m  B )  =  ( (/)  ^m  B ) )
20 map0b 6806 . . 3  |-  ( B  =/=  (/)  ->  ( (/)  ^m  B
)  =  (/) )
2119, 20sylan9eq 2335 . 2  |-  ( ( A  =  (/)  /\  B  =/=  (/) )  ->  ( A  ^m  B )  =  (/) )
2218, 21impbid1 194 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   {csn 3640    X. cxp 4687   -->wf 5251  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  map0  6808  mapdom2  7032
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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