Theorem map0 4344

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 4332	. . . . 5 |- (A ^m B) = {f | f:B-->A}
4	3	eqeq1i 1482	. . . 4 |- ((A ^m B) = (/) <-> {f | f:B-->A} = (/))
5		snssi 2466	. . . . . . . 8 |- (x e. A -> {x} (_ A)
6		visset 1813	. . . . . . . . . 10 |- x e. V
7	6	fconst 3658	. . . . . . . . 9 |- (B X. {x}):B-->{x}
8		fss 3635	. . . . . . . . 9 |- (((B X. {x}):B-->{x} /\ {x} (_ A) -> (B X. {x}):B-->A)
9	7, 8	mpan 695	. . . . . . . 8 |- ({x} (_ A -> (B X. {x}):B-->A)
10		snex 2750	. . . . . . . . . 10 |- {x} e. V
11	2, 10	xpex 3260	. . . . . . . . 9 |- (B X. {x}) e. V
12		feq1 3620	. . . . . . . . 9 |- (f = (B X. {x}) -> (f:B-->A <-> (B X. {x}):B-->A))
13	11, 12	cla4ev 1869	. . . . . . . 8 |- ((B X. {x}):B-->A -> E.f f:B-->A)
14	5, 9, 13	3syl 20	. . . . . . 7 |- (x e. A -> E.f f:B-->A)
15	14	19.23aiv 1295	. . . . . 6 |- (E.x x e. A -> E.f f:B-->A)
16		ne0 2288	. . . . . 6 |- (A =/= (/) <-> E.x x e. A)
17		abn0 2290	. . . . . 6 |- ({f | f:B-->A} =/= (/) <-> E.f f:B-->A)
18	15, 16, 17	3imtr4 219	. . . . 5 |- (A =/= (/) -> {f | f:B-->A} =/= (/))
19	18	necon4i 1625	. . . 4 |- ({f | f:B-->A} = (/) -> A = (/))
20	4, 19	sylbi 199	. . 3 |- ((A ^m B) = (/) -> A = (/))
21		0ex 2711	. . . . . . 7 |- (/) e. V
22	21	snnz 2458	. . . . . 6 |- {(/)} =/= (/)
23	1	map0e 4342	. . . . . . . 8 |- (A ^m (/)) = 1o
24		df1o2 4140	. . . . . . . 8 |- 1o = {(/)}
25	23, 24	eqtr 1495	. . . . . . 7 |- (A ^m (/)) = {(/)}
26	25	neeq1i 1592	. . . . . 6 |- ((A ^m (/)) =/= (/) <-> {(/)} =/= (/))
27	22, 26	mpbir 190	. . . . 5 |- (A ^m (/)) =/= (/)
28		opreq2 3969	. . . . . 6 |- (B = (/) -> (A ^m B) = (A ^m (/)))
29	28	neeq1d 1594	. . . . 5 |- (B = (/) -> ((A ^m B) =/= (/) <-> (A ^m (/)) =/= (/)))
30	27, 29	mpbiri 194	. . . 4 |- (B = (/) -> (A ^m B) =/= (/))
31	30	necon2i 1613	. . 3 |- ((A ^m B) = (/) -> B =/= (/))
32	20, 31	jca 288	. 2 |- ((A ^m B) = (/) -> (A = (/) /\ B =/= (/)))
33		opreq1 3968	. . 3 |- (A = (/) -> (A ^m B) = ((/) ^m B))
34	2	map0b 4343	. . 3 |- (B =/= (/) -> ((/) ^m B) = (/))
35	33, 34	sylan9eq 1527	. 2 |- ((A = (/) /\ B =/= (/)) -> (A ^m B) = (/))
36	32, 35	impbi 157	1 |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  Vcvv 1811   (_ wss 2047  (/)c0 2280  {csn 2409   X. cxp 3168  -->wf 3178  (class class class)co 3963  1oc1o 4128   ^m cm 4322

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-1o 4133  df-map 4324

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