Theorem map0b 6665

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 6648	. . . 4 |- ( f e. ( (/) ^m A ) -> f : A --> (/) )
2		fdm 5353	. . . . 5 |- ( f : A --> (/) -> dom f = A )
3		frn 5356	. . . . . . 7 |- ( f : A --> (/) -> ran f C_ (/) )
4		ss0 3455	. . . . . . 7 |- ( ran f C_ (/) -> ran f = (/) )
5	3, 4	syl 14	. . . . . 6 |- ( f : A --> (/) -> ran f = (/) )
6		dm0rn0 4828	. . . . . 6 |- ( dom f = (/) <-> ran f = (/) )
7	5, 6	sylibr 133	. . . . 5 |- ( f : A --> (/) -> dom f = (/) )
8	2, 7	eqtr3d 2205	. . . 4 |- ( f : A --> (/) -> A = (/) )
9	1, 8	syl 14	. . 3 |- ( f e. ( (/) ^m A ) -> A = (/) )
10	9	necon3ai 2389	. 2 |- ( A =/= (/) -> -. f e. ( (/) ^m A ) )
11	10	eq0rdv 3459	1 |- ( A =/= (/) -> ( (/) ^m A ) = (/) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   =/= wne 2340   C_ wss 3121   (/)c0 3414   dom cdm 4611   ran crn 4612   -->wf 5194  (class class class)co 5853   ^m cmap 6626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628

This theorem is referenced by:  map0g  6666