Theorem map0b 6024

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 4110	. . 3 |- (/) e. _V
2		map0e.1	. . 3 |- A e. _V
3	1, 2	mapval 6011	. 2 |- ((/) ^m A) = {f | f:A-->(/)}
4		abn0 3568	. . . 4 |- ({f | f:A-->(/)} =/= (/) <-> E.f f:A-->(/))
5		fdm 5226	. . . . . 6 |- (f:A-->(/) -> dom f = A)
6		frn 5228	. . . . . . . 8 |- (f:A-->(/) -> ran f C_ (/))
7		ss0 3581	. . . . . . . 8 |- (ran f C_ (/) -> ran f = (/))
8	6, 7	syl 15	. . . . . . 7 |- (f:A-->(/) -> ran f = (/))
9		dm0rn0 4921	. . . . . . 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 2560	. 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 2516  _Vcvv 2859   C_ wss 3257  (/)c0 3550  dom cdm 4772  ran crn 4773  -->wf 4777  (class class class)co 5525   ^m cmap 5999

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-map 6001

This theorem is referenced by:  map0  6025