Theorem relsn2m 5205

Description: A singleton is a relation iff it has an inhabited domain. (Contributed by Jim Kingdon, 16-Dec-2018.)

Hypothesis

Ref	Expression
relsn2m.1	|- A e. _V

Assertion

Ref	Expression
relsn2m	|- ( Rel { A } <-> E. x x e. dom { A } )

Distinct variable group:   x, A

Proof of Theorem relsn2m

Step	Hyp	Ref	Expression
1		relsn2m.1	. . 3 |- A e. _V
2	1	relsn 4829	. 2 |- ( Rel { A } <-> A e. ( _V X. _V ) )
3		dmsnm 5200	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
4	2, 3	bitri 184	1 |- ( Rel { A } <-> E. x x e. dom { A } )

Syntax hints:   <-> wb 105   E.wex 1538   e. wcel 2200   _Vcvv 2800   {csn 3667   X. cxp 4721   dom cdm 4723   Rel wrel 4728

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-dm 4733

This theorem is referenced by: (None)