Theorem mapdm0OLD 39383

Description: The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
mapdm0OLD	|- ( A e. V -> ( A ^m (/) ) = { (/) } )

Proof of Theorem mapdm0OLD

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. . . . . 6 |- (/) e. _V
2		elmapg 7870	. . . . . 6 |- ( ( A e. V /\ (/) e. _V ) -> ( f e. ( A ^m (/) ) <-> f : (/) --> A ) )
3	1, 2	mpan2 707	. . . . 5 |- ( A e. V -> ( f e. ( A ^m (/) ) <-> f : (/) --> A ) )
4	3	biimpa 501	. . . 4 |- ( ( A e. V /\ f e. ( A ^m (/) ) ) -> f : (/) --> A )
5		f0bi 6088	. . . 4 |- ( f : (/) --> A <-> f = (/) )
6	4, 5	sylib 208	. . 3 |- ( ( A e. V /\ f e. ( A ^m (/) ) ) -> f = (/) )
7	6	ralrimiva 2966	. 2 |- ( A e. V -> A. f e. ( A ^m (/) ) f = (/) )
8		f0 6086	. . . . . 6 |- (/) : (/) --> A
9	8	a1i 11	. . . . 5 |- ( A e. V -> (/) : (/) --> A )
10		id 22	. . . . . 6 |- ( A e. V -> A e. V )
11	1	a1i 11	. . . . . 6 |- ( A e. V -> (/) e. _V )
12		elmapg 7870	. . . . . 6 |- ( ( A e. V /\ (/) e. _V ) -> ( (/) e. ( A ^m (/) ) <-> (/) : (/) --> A ) )
13	10, 11, 12	syl2anc 693	. . . . 5 |- ( A e. V -> ( (/) e. ( A ^m (/) ) <-> (/) : (/) --> A ) )
14	9, 13	mpbird 247	. . . 4 |- ( A e. V -> (/) e. ( A ^m (/) ) )
15		ne0i 3921	. . . 4 |- ( (/) e. ( A ^m (/) ) -> ( A ^m (/) ) =/= (/) )
16	14, 15	syl 17	. . 3 |- ( A e. V -> ( A ^m (/) ) =/= (/) )
17		eqsn 4361	. . 3 |- ( ( A ^m (/) ) =/= (/) -> ( ( A ^m (/) ) = { (/) } <-> A. f e. ( A ^m (/) ) f = (/) ) )
18	16, 17	syl 17	. 2 |- ( A e. V -> ( ( A ^m (/) ) = { (/) } <-> A. f e. ( A ^m (/) ) f = (/) ) )
19	7, 18	mpbird 247	1 |- ( A e. V -> ( A ^m (/) ) = { (/) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   {csn 4177   -->wf 5884  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by: (None)