ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  map0g Unicode version

Theorem map0g 6713
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 fconst6g 5433 . . . . . . . 8  |-  ( f  e.  A  ->  ( B  X.  { f } ) : B --> A )
2 elmapg 6686 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( B  X.  { f } )  e.  ( A  ^m  B )  <->  ( B  X.  { f } ) : B --> A ) )
31, 2imbitrrid 156 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( B  X.  {
f } )  e.  ( A  ^m  B
) ) )
4 ne0i 3444 . . . . . . 7  |-  ( ( B  X.  { f } )  e.  ( A  ^m  B )  ->  ( A  ^m  B )  =/=  (/) )
53, 4syl6 33 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( f  e.  A  ->  ( A  ^m  B
)  =/=  (/) ) )
65exlimdv 1830 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( E. f  f  e.  A  ->  ( A  ^m  B )  =/=  (/) ) )
76necon2bd 2418 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  -.  E. f  f  e.  A ) )
8 notm0 3458 . . . 4  |-  ( -. 
E. f  f  e.  A  <->  A  =  (/) )
97, 8imbitrdi 161 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  A  =  (/) ) )
10 f0 5425 . . . . . . 7  |-  (/) : (/) --> A
11 feq2 5368 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/) : B --> A  <->  (/) : (/) --> A ) )
1210, 11mpbiri 168 . . . . . 6  |-  ( B  =  (/)  ->  (/) : B --> A )
13 elmapg 6686 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( (/)  e.  ( A  ^m  B )  <->  (/) : B --> A ) )
1412, 13imbitrrid 156 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  -> 
(/)  e.  ( A  ^m  B ) ) )
15 ne0i 3444 . . . . 5  |-  ( (/)  e.  ( A  ^m  B
)  ->  ( A  ^m  B )  =/=  (/) )
1614, 15syl6 33 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  =  (/)  ->  ( A  ^m  B
)  =/=  (/) ) )
1716necon2d 2419 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  B  =/=  (/) ) )
189, 17jcad 307 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  ->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
19 oveq1 5902 . . 3  |-  ( A  =  (/)  ->  ( A  ^m  B )  =  ( (/)  ^m  B ) )
20 map0b 6712 . . 3  |-  ( B  =/=  (/)  ->  ( (/)  ^m  B
)  =  (/) )
2119, 20sylan9eq 2242 . 2  |-  ( ( A  =  (/)  /\  B  =/=  (/) )  ->  ( A  ^m  B )  =  (/) )
2218, 21impbid1 142 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  ^m  B )  =  (/)  <->  ( A  =  (/)  /\  B  =/=  (/) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1364   E.wex 1503    e. wcel 2160    =/= wne 2360   (/)c0 3437   {csn 3607    X. cxp 4642   -->wf 5231  (class class class)co 5895    ^m cmap 6673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-map 6675
This theorem is referenced by:  map0  6714
  Copyright terms: Public domain W3C validator