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Theorem map0g 6693
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
StepHypRef Expression
1 n0 3371 . . . . 5  |-  ( A  =/=  (/)  <->  E. f  f  e.  A )
2 fconst6g 5287 . . . . . . . 8  |-  ( f  e.  A  ->  ( B  X.  { f } ) : B --> A )
3 elmapg 6671 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( B  X.  { f } )  e.  ( A  ^m  B )  <->  ( B  X.  { f } ) : B --> A ) )
42, 3syl5ibr 214 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( B  X.  {
f } )  e.  ( A  ^m  B
) ) )
5 ne0i 3368 . . . . . . 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 1932 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. f  f  e.  A  ->  ( A  ^m  B )  =/=  (/) ) )
81, 7syl5bi 210 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  =/=  (/)  ->  ( A  ^m  B )  =/=  (/) ) )
98necon4d 2475 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  A  =  (/) ) )
10 f0 5282 . . . . . . 7  |-  (/) : (/) --> A
11 feq2 5233 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/) : B --> A  <->  (/) : (/) --> A ) )
1210, 11mpbiri 226 . . . . . 6  |-  ( B  =  (/)  ->  (/) : B --> A )
13 elmapg 6671 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (/)  e.  ( A  ^m  B )  <->  (/) : B --> A ) )
1412, 13syl5ibr 214 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  -> 
(/)  e.  ( A  ^m  B ) ) )
15 ne0i 3368 . . . . 5  |-  ( (/)  e.  ( A  ^m  B
)  ->  ( A  ^m  B )  =/=  (/) )
1614, 15syl6 31 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  ->  ( A  ^m  B
)  =/=  (/) ) )
1716necon2d 2462 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  B  =/=  (/) ) )
189, 17jcad 521 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
19 oveq1 5717 . . 3  |-  ( A  =  (/)  ->  ( A  ^m  B )  =  ( (/)  ^m  B ) )
20 map0b 6692 . . 3  |-  ( B  =/=  (/)  ->  ( (/)  ^m  B
)  =  (/) )
2119, 20sylan9eq 2305 . 2  |-  ( ( A  =  (/)  /\  B  =/=  (/) )  ->  ( A  ^m  B )  =  (/) )
2218, 21impbid1 196 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621    =/= wne 2412   (/)c0 3362   {csn 3544    X. cxp 4578   -->wf 4588  (class class class)co 5710    ^m cmap 6658
This theorem is referenced by:  map0  6694  mapdom2  6917
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-map 6660
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