Theorem map0b 6025

Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by set.mm contributors, 10-Dec-2003.) (Revised by set.mm contributors, 19-Mar-2007.)

Hypothesis

Ref	Expression
map0e.1	|- A e. _V

Assertion

Ref	Expression
map0b	|- (A =/= (/) -> ((/) ^m A) = (/))

Proof of Theorem map0b

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4111	. . 3 |- (/) e. _V
2		map0e.1	. . 3 |- A e. _V
3	1, 2	mapval 6012	. 2 |- ((/) ^m A) = {f | f:A-->(/)}
4		abn0 3569	. . . 4 |- ({f | f:A-->(/)} =/= (/) <-> E.f f:A-->(/))
5		fdm 5227	. . . . . 6 |- (f:A-->(/) -> dom f = A)
6		frn 5229	. . . . . . . 8 |- (f:A-->(/) -> ran f C_ (/))
7		ss0 3582	. . . . . . . 8 |- (ran f C_ (/) -> ran f = (/))
8	6, 7	syl 15	. . . . . . 7 |- (f:A-->(/) -> ran f = (/))
9		dm0rn0 4922	. . . . . . 7 |- (dom f = (/) <-> ran f = (/))
10	8, 9	sylibr 203	. . . . . 6 |- (f:A-->(/) -> dom f = (/))
11	5, 10	eqtr3d 2387	. . . . 5 |- (f:A-->(/) -> A = (/))
12	11	exlimiv 1634	. . . 4 |- (E.f f:A-->(/) -> A = (/))
13	4, 12	sylbi 187	. . 3 |- ({f | f:A-->(/)} =/= (/) -> A = (/))
14	13	necon1i 2561	. 2 |- (A =/= (/) -> {f | f:A-->(/)} = (/))
15	3, 14	syl5eq 2397	1 |- (A =/= (/) -> ((/) ^m A) = (/))

Syntax hints:   -> wi 4  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339   =/= wne 2517  _Vcvv 2860   C_ wss 3258  (/)c0 3551  dom cdm 4773  ran crn 4774  -->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:  map0  6026