Theorem rnsnm 5133

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 5132	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
2		dmmrnm 4882	. 2 |- ( E. x x e. dom { A } <-> E. x x e. ran { A } )
3	1, 2	bitri 184	1 |- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } )

Syntax hints:   <-> wb 105   E.wex 1503   e. wcel 2164   _Vcvv 2760   {csn 3619   X. cxp 4658   dom cdm 4660   ran crn 4661

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by: (None)