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Theorem map0g 6740
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 3406 . . . . 5  |-  ( A  =/=  (/)  <->  E. f  f  e.  A )
2 fconst6g 5333 . . . . . . . 8  |-  ( f  e.  A  ->  ( B  X.  { f } ) : B --> A )
3 elmapg 6718 . . . . . . . 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 3403 . . . . . . 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 1933 . . . . 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 2482 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  A  =  (/) ) )
10 f0 5328 . . . . . . 7  |-  (/) : (/) --> A
11 feq2 5279 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/) : B --> A  <->  (/) : (/) --> A ) )
1210, 11mpbiri 226 . . . . . 6  |-  ( B  =  (/)  ->  (/) : B --> A )
13 elmapg 6718 . . . . . 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 3403 . . . . 5  |-  ( (/)  e.  ( A  ^m  B
)  ->  ( A  ^m  B )  =/=  (/) )
1614, 15syl6 31 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  ->  ( A  ^m  B
)  =/=  (/) ) )
1716necon2d 2469 . . 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 5764 . . 3  |-  ( A  =  (/)  ->  ( A  ^m  B )  =  ( (/)  ^m  B ) )
20 map0b 6739 . . 3  |-  ( B  =/=  (/)  ->  ( (/)  ^m  B
)  =  (/) )
2119, 20sylan9eq 2308 . 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 2419   (/)c0 3397   {csn 3581    X. cxp 4624   -->wf 4634  (class class class)co 5757    ^m cmap 6705
This theorem is referenced by:  map0  6741  mapdom2  6965
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 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-map 6707
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