Theorem map0b 6921

Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)

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		elmapi 6904	. . . 4 |- ( f e. ( (/) ^m A ) -> f : A --> (/) )
2		fdm 5514	. . . . 5 |- ( f : A --> (/) -> dom f = A )
3		frn 5517	. . . . . . 7 |- ( f : A --> (/) -> ran f C_ (/) )
4		ss0 3549	. . . . . . 7 |- ( ran f C_ (/) -> ran f = (/) )
5	3, 4	syl 14	. . . . . 6 |- ( f : A --> (/) -> ran f = (/) )
6		dm0rn0 4973	. . . . . 6 |- ( dom f = (/) <-> ran f = (/) )
7	5, 6	sylibr 134	. . . . 5 |- ( f : A --> (/) -> dom f = (/) )
8	2, 7	eqtr3d 2267	. . . 4 |- ( f : A --> (/) -> A = (/) )
9	1, 8	syl 14	. . 3 |- ( f e. ( (/) ^m A ) -> A = (/) )
10	9	necon3ai 2461	. 2 |- ( A =/= (/) -> -. f e. ( (/) ^m A ) )
11	10	eq0rdv 3553	1 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   =/= wne 2412   C_ wss 3211   (/)c0 3508   dom cdm 4749   ran crn 4750   -->wf 5348  (class class class)co 6050   ^m cmap 6882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-map 6884

This theorem is referenced by:  map0g  6922