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Theorem rnsnop 5351
Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)
Hypothesis
Ref Expression
cnvsn.1  |-  A  e. 
_V
Assertion
Ref Expression
rnsnop  |-  ran  { <. A ,  B >. }  =  { B }

Proof of Theorem rnsnop
StepHypRef Expression
1 cnvsn.1 . 2  |-  A  e. 
_V
2 rnsnopg 5350 . 2  |-  ( A  e.  _V  ->  ran  {
<. A ,  B >. }  =  { B }
)
31, 2ax-mp 8 1  |-  ran  { <. A ,  B >. }  =  { B }
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   _Vcvv 2957   {csn 3815   <.cop 3818   ran crn 4880
This theorem is referenced by:  op2nda  5355  fpr  5915  en1  7175  fodomfi  7386  dcomex  8328  ex-rn  21749  ex-ima  21751  gidsn  21937  ginvsn  21938  rngosn  21993  zrdivrng  22021  ghomsn  25100  axlowdimlem13  25894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890
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