Theorem rnsnm 5064

Description: The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)

Assertion

Ref	Expression
rnsnm	|- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } )

Distinct variable group:   x, A

Proof of Theorem rnsnm

Step	Hyp	Ref	Expression
1		dmsnm 5063	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
2		dmmrnm 4817	. 2 |- ( E. x x e. dom { A } <-> E. x x e. ran { A } )
3	1, 2	bitri 183	1 |- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } )

Syntax hints:   <-> wb 104   E.wex 1479   e. wcel 2135   _Vcvv 2721   {csn 3570   X. cxp 4596   dom cdm 4598   ran crn 4599

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-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2723  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-br 3977  df-opab 4038  df-xp 4604  df-cnv 4606  df-dm 4608  df-rn 4609

This theorem is referenced by: (None)