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Theorem map0 4328
Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89.
Hypotheses
Ref Expression
map0.1 |- A e. V
map0.2 |- B e. V
Assertion
Ref Expression
map0 |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))
Proof of Theorem map0
StepHypRef Expression
1 map0.1 . . . . . 6 |- A e. V
2 map0.2 . . . . . 6 |- B e. V
31, 2mapval 4316 . . . . 5 |- (A ^m B) = {f | f:B-->A}
43eqeq1i 1474 . . . 4 |- ((A ^m B) = (/) <-> {f | f:B-->A} = (/))
5 snssi 2457 . . . . . . . 8 |- (x e. A -> {x} (_ A)
6 visset 1804 . . . . . . . . . 10 |- x e. V
76fconst 3643 . . . . . . . . 9 |- (B X. {x}):B-->{x}
8 fss 3620 . . . . . . . . 9 |- (((B X. {x}):B-->{x} /\ {x} (_ A) -> (B X. {x}):B-->A)
97, 8mpan 693 . . . . . . . 8 |- ({x} (_ A -> (B X. {x}):B-->A)
10 snex 2740 . . . . . . . . . 10 |- {x} e. V
112, 10xpex 3250 . . . . . . . . 9 |- (B X. {x}) e. V
12 feq1 3606 . . . . . . . . 9 |- (f = (B X. {x}) -> (f:B-->A <-> (B X. {x}):B-->A))
1311, 12cla4ev 1860 . . . . . . . 8 |- ((B X. {x}):B-->A -> E.f f:B-->A)
145, 9, 133syl 20 . . . . . . 7 |- (x e. A -> E.f f:B-->A)
151419.23aiv 1290 . . . . . 6 |- (E.x x e. A -> E.f f:B-->A)
16 ne0 2278 . . . . . 6 |- (A =/= (/) <-> E.x x e. A)
17 abn0 2280 . . . . . 6 |- ({f | f:B-->A} =/= (/) <-> E.f f:B-->A)
1815, 16, 173imtr4 219 . . . . 5 |- (A =/= (/) -> {f | f:B-->A} =/= (/))
1918necon4i 1617 . . . 4 |- ({f | f:B-->A} = (/) -> A = (/))
204, 19sylbi 199 . . 3 |- ((A ^m B) = (/) -> A = (/))
21 0ex 2701 . . . . . . 7 |- (/) e. V
2221snnz 2449 . . . . . 6 |- {(/)} =/= (/)
231map0e 4326 . . . . . . . 8 |- (A ^m (/)) = 1o
24 df1o2 4124 . . . . . . . 8 |- 1o = {(/)}
2523, 24eqtr 1487 . . . . . . 7 |- (A ^m (/)) = {(/)}
2625neeq1i 1584 . . . . . 6 |- ((A ^m (/)) =/= (/) <-> {(/)} =/= (/))
2722, 26mpbir 190 . . . . 5 |- (A ^m (/)) =/= (/)
28 opreq2 3954 . . . . . 6 |- (B = (/) -> (A ^m B) = (A ^m (/)))
2928neeq1d 1586 . . . . 5 |- (B = (/) -> ((A ^m B) =/= (/) <-> (A ^m (/)) =/= (/)))
3027, 29mpbiri 194 . . . 4 |- (B = (/) -> (A ^m B) =/= (/))
3130necon2i 1605 . . 3 |- ((A ^m B) = (/) -> B =/= (/))
3220, 31jca 288 . 2 |- ((A ^m B) = (/) -> (A = (/) /\ B =/= (/)))
33 opreq1 3953 . . 3 |- (A = (/) -> (A ^m B) = ((/) ^m B))
342map0b 4327 . . 3 |- (B =/= (/) -> ((/) ^m B) = (/))
3533, 34sylan9eq 1519 . 2 |- ((A = (/) /\ B =/= (/)) -> (A ^m B) = (/))
3632, 35impbi 157 1 |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456   =/= wne 1577  Vcvv 1802   (_ wss 2037  (/)c0 2270  {csn 2399   X. cxp 3158  -->wf 3168  (class class class)co 3948  1oc1o 4112   ^m cm 4306
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-map 4308
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