Theorem map0 5403

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 5391	. . . . 5 |- (A ^m B) = {f | f:B-->A}
4	3	eqeq1i 1891	. . . 4 |- ((A ^m B) = (/) <-> {f | f:B-->A} = (/))
5		snssi 3129	. . . . . . . 8 |- (x e. A -> {x} C_ A)
6		visset 2295	. . . . . . . . . 10 |- x e. _V
7	6	fconst 4602	. . . . . . . . 9 |- (B X. {x}):B-->{x}
8		fss 4571	. . . . . . . . 9 |- (((B X. {x}):B-->{x} /\ {x} C_ A) -> (B X. {x}):B-->A)
9	7, 8	mpan 759	. . . . . . . 8 |- ({x} C_ A -> (B X. {x}):B-->A)
10		snex 3492	. . . . . . . . . 10 |- {x} e. _V
11	2, 10	xpex 4096	. . . . . . . . 9 |- (B X. {x}) e. _V
12		feq1 4551	. . . . . . . . 9 |- (f = (B X. {x}) -> (f:B-->A <-> (B X. {x}):B-->A))
13	11, 12	cla4ev 2371	. . . . . . . 8 |- ((B X. {x}):B-->A -> E.f f:B-->A)
14	5, 9, 13	3syl 24	. . . . . . 7 |- (x e. A -> E.f f:B-->A)
15	14	19.23aiv 1674	. . . . . 6 |- (E.x x e. A -> E.f f:B-->A)
16		n0 2884	. . . . . 6 |- (A =/= (/) <-> E.x x e. A)
17		abn0 2892	. . . . . 6 |- ({f | f:B-->A} =/= (/) <-> E.f f:B-->A)
18	15, 16, 17	3imtr4i 236	. . . . 5 |- (A =/= (/) -> {f | f:B-->A} =/= (/))
19	18	necon4i 2069	. . . 4 |- ({f | f:B-->A} = (/) -> A = (/))
20	4, 19	sylbi 216	. . 3 |- ((A ^m B) = (/) -> A = (/))
21		0ex 3446	. . . . . . 7 |- (/) e. _V
22	21	snnz 3119	. . . . . 6 |- {(/)} =/= (/)
23	1	map0e 5401	. . . . . . . 8 |- (A ^m (/)) = 1o
24		df1o2 5185	. . . . . . . 8 |- 1o = {(/)}
25	23, 24	eqtri 1908	. . . . . . 7 |- (A ^m (/)) = {(/)}
26	25	neeq1i 2026	. . . . . 6 |- ((A ^m (/)) =/= (/) <-> {(/)} =/= (/))
27	22, 26	mpbir 207	. . . . 5 |- (A ^m (/)) =/= (/)
28		opreq2 4890	. . . . . 6 |- (B = (/) -> (A ^m B) = (A ^m (/)))
29	28	neeq1d 2028	. . . . 5 |- (B = (/) -> ((A ^m B) =/= (/) <-> (A ^m (/)) =/= (/)))
30	27, 29	mpbiri 211	. . . 4 |- (B = (/) -> (A ^m B) =/= (/))
31	30	necon2i 2054	. . 3 |- ((A ^m B) = (/) -> B =/= (/))
32	20, 31	jca 310	. 2 |- ((A ^m B) = (/) -> (A = (/) /\ B =/= (/)))
33		opreq1 4889	. . 3 |- (A = (/) -> (A ^m B) = ((/) ^m B))
34	2	map0b 5402	. . 3 |- (B =/= (/) -> ((/) ^m B) = (/))
35	33, 34	sylan9eq 1948	. 2 |- ((A = (/) /\ B =/= (/)) -> (A ^m B) = (/))
36	32, 35	impbii 174	1 |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))

Syntax hints:   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044   X. cxp 3984  -->wf 3994  (class class class)co 4884  1oc1o 5172   ^m cmap 5381

This theorem is referenced by:  rrndm 16015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1o 5177  df-map 5383

Copyright terms: Public domain