Theorem map0 3268

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

Step	Hyp	Ref	Expression
1		map0.1	. . . . . 6 |- A e. V
2		map0.2	. . . . . 6 |- B e. V
3	1, 2	mapval 3264	. . . . 5 |- (A ^m B) = {f | f:B-->A}
4	3	cleq1i 1108	. . . 4 |- ((A ^m B) = (/) <-> {f | f:B-->A} = (/))
5		snssi 1851	. . . . . . . 8 |- (x e. A -> {x} (_ A)
6		visset 1350	. . . . . . . . . 10 |- x e. V
7	6	fconst 2774	. . . . . . . . 9 |- (B X. {x}):B-->{x}
8		fss 2759	. . . . . . . . 9 |- (((B X. {x}):B-->{x} /\ {x} (_ A) -> (B X. {x}):B-->A)
9	7, 8	mpan 518	. . . . . . . 8 |- ({x} (_ A -> (B X. {x}):B-->A)
10		snex 1859	. . . . . . . . . 10 |- {x} e. V
11	2, 10	xpex 2488	. . . . . . . . 9 |- (B X. {x}) e. V
12		feq1 2748	. . . . . . . . 9 |- (f = (B X. {x}) -> (f:B-->A <-> (B X. {x}):B-->A))
13	11, 12	cla4ev 1401	. . . . . . . 8 |- ((B X. {x}):B-->A -> E.f f:B-->A)
14	5, 9, 13	3syl 21	. . . . . . 7 |- (x e. A -> E.f f:B-->A)
15	14	19.23aiv 952	. . . . . 6 |- (E.x x e. A -> E.f f:B-->A)
16		n0 1714	. . . . . 6 |- (-. A = (/) <-> E.x x e. A)
17		abn0 1715	. . . . . 6 |- (-. {f | f:B-->A} = (/) <-> E.f f:B-->A)
18	15, 16, 17	3imtr4 192	. . . . 5 |- (-. A = (/) -> -. {f | f:B-->A} = (/))
19	18	a3i 69	. . . 4 |- ({f | f:B-->A} = (/) -> A = (/))
20	4, 19	sylbi 174	. . 3 |- ((A ^m B) = (/) -> A = (/))
21		0nep0 1887	. . . . . 6 |- -. (/) = {(/)}
22	1	map0e 3266	. . . . . . . 8 |- (A ^m (/)) = 1o
23	22	cleq1i 1108	. . . . . . 7 |- ((A ^m (/)) = (/) <-> 1o = (/))
24		df1o2 3111	. . . . . . . 8 |- 1o = {(/)}
25	24	cleq1i 1108	. . . . . . 7 |- (1o = (/) <-> {(/)} = (/))
26		cleqcom 1103	. . . . . . 7 |- ({(/)} = (/) <-> (/) = {(/)})
27	23, 25, 26	3bitr 155	. . . . . 6 |- ((A ^m (/)) = (/) <-> (/) = {(/)})
28	21, 27	mtbir 167	. . . . 5 |- -. (A ^m (/)) = (/)
29		opreq2 3007	. . . . . 6 |- (B = (/) -> (A ^m B) = (A ^m (/)))
30	29	cleq1d 1109	. . . . 5 |- (B = (/) -> ((A ^m B) = (/) <-> (A ^m (/)) = (/)))
31	28, 30	mtbiri 539	. . . 4 |- (B = (/) -> -. (A ^m B) = (/))
32	31	con2i 89	. . 3 |- ((A ^m B) = (/) -> -. B = (/))
33	20, 32	jca 236	. 2 |- ((A ^m B) = (/) -> (A = (/) /\ -. B = (/)))
34		opreq1 3006	. . 3 |- (A = (/) -> (A ^m B) = ((/) ^m B))
35	2	map0b 3267	. . 3 |- (-. B = (/) -> ((/) ^m B) = (/))
36	34, 35	sylan9eq 1144	. 2 |- ((A = (/) /\ -. B = (/)) -> (A ^m B) = (/))
37	33, 36	impbi 139	1 |- ((A ^m B) = (/) <-> (A = (/) /\ -. B = (/)))

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  (/)c0 1707  {csn 1808   X. cxp 2408  -->wf 2418  (class class class)co 3001  1oc1o 3099   ^m cm 3258

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-oprab 3004  df-1o 3104  df-map 3259

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