Theorem rnsnm 4787

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 4786	. 2 |- ( A e. ( _V X. _V ) <-> E. x x e. dom { A } )
2		dmmrnm 4554	. 2 |- ( E. x x e. dom { A } <-> E. x x e. ran { A } )
3	1, 2	bitri 173	1 |- ( A e. ( _V X. _V ) <-> E. x x e. ran { A } )

Syntax hints:   <-> wb 98   E.wex 1381   e. wcel 1393   _Vcvv 2557   {csn 3375   X. cxp 4343   dom cdm 4345   ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by: (None)