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Theorem rnsnm 4837
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
StepHypRef Expression
1 dmsnm 4836 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
dom  { A } )
2 dmmrnm 4602 . 2  |-  ( E. x  x  e.  dom  { A }  <->  E. x  x  e.  ran  { A } )
31, 2bitri 182 1  |-  ( A  e.  ( _V  X.  _V )  <->  E. x  x  e. 
ran  { A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   E.wex 1422    e. wcel 1434   _Vcvv 2610   {csn 3416    X. cxp 4389   dom cdm 4391   ran crn 4392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-cnv 4399  df-dm 4401  df-rn 4402
This theorem is referenced by: (None)
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