Theorem map0 6026

Description: Set exponentiation is empty iff the base is empty and the exponent is not empty. Theorem 97 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 17-May-2007.)

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

Dummy variables x f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		map0.1	. . . . . 6 |- A e. _V
2		map0.2	. . . . . 6 |- B e. _V
3	1, 2	mapval 6012	. . . . 5 |- (A ^m B) = {f | f:B-->A}
4	3	eqeq1i 2360	. . . 4 |- ((A ^m B) = (/) <-> {f | f:B-->A} = (/))
5		snssi 3853	. . . . . . . 8 |- (x e. A -> {x} C_ A)
6		vex 2863	. . . . . . . . . 10 |- x e. _V
7	6	fconst 5251	. . . . . . . . 9 |- (B X. {x}):B-->{x}
8		fss 5231	. . . . . . . . 9 |- (((B X. {x}):B-->{x} /\ {x} C_ A) -> (B X. {x}):B-->A)
9	7, 8	mpan 651	. . . . . . . 8 |- ({x} C_ A -> (B X. {x}):B-->A)
10		snex 4112	. . . . . . . . . 10 |- {x} e. _V
11	2, 10	xpex 5116	. . . . . . . . 9 |- (B X. {x}) e. _V
12		feq1 5211	. . . . . . . . 9 |- (f = (B X. {x}) -> (f:B-->A <-> (B X. {x}):B-->A))
13	11, 12	spcev 2947	. . . . . . . 8 |- ((B X. {x}):B-->A -> E.f f:B-->A)
14	5, 9, 13	3syl 18	. . . . . . 7 |- (x e. A -> E.f f:B-->A)
15	14	exlimiv 1634	. . . . . 6 |- (E.x x e. A -> E.f f:B-->A)
16		n0 3560	. . . . . 6 |- (A =/= (/) <-> E.x x e. A)
17		abn0 3569	. . . . . 6 |- ({f | f:B-->A} =/= (/) <-> E.f f:B-->A)
18	15, 16, 17	3imtr4i 257	. . . . 5 |- (A =/= (/) -> {f | f:B-->A} =/= (/))
19	18	necon4i 2577	. . . 4 |- ({f | f:B-->A} = (/) -> A = (/))
20	4, 19	sylbi 187	. . 3 |- ((A ^m B) = (/) -> A = (/))
21	1	map0e 6024	. . . . . 6 |- (A ^m (/)) = {(/)}
22		0ex 4111	. . . . . . . 8 |- (/) e. _V
23	22	snid 3761	. . . . . . 7 |- (/) e. {(/)}
24		ne0i 3557	. . . . . . 7 |- ((/) e. {(/)} -> {(/)} =/= (/))
25	23, 24	ax-mp 5	. . . . . 6 |- {(/)} =/= (/)
26	21, 25	eqnetri 2534	. . . . 5 |- (A ^m (/)) =/= (/)
27		oveq2 5532	. . . . . 6 |- (B = (/) -> (A ^m B) = (A ^m (/)))
28	27	neeq1d 2530	. . . . 5 |- (B = (/) -> ((A ^m B) =/= (/) <-> (A ^m (/)) =/= (/)))
29	26, 28	mpbiri 224	. . . 4 |- (B = (/) -> (A ^m B) =/= (/))
30	29	necon2i 2564	. . 3 |- ((A ^m B) = (/) -> B =/= (/))
31	20, 30	jca 518	. 2 |- ((A ^m B) = (/) -> (A = (/) /\ B =/= (/)))
32		oveq1 5531	. . 3 |- (A = (/) -> (A ^m B) = ((/) ^m B))
33	2	map0b 6025	. . 3 |- (B =/= (/) -> ((/) ^m B) = (/))
34	32, 33	sylan9eq 2405	. 2 |- ((A = (/) /\ B =/= (/)) -> (A ^m B) = (/))
35	31, 34	impbii 180	1 |- ((A ^m B) = (/) <-> (A = (/) /\ B =/= (/)))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  _Vcvv 2860   C_ wss 3258  (/)c0 3551  {csn 3738   X. cxp 4771  -->wf 4778  (class class class)co 5526   ^m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by: (None)