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Theorem rnsnop 5337
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 5335 . 2  |-  ( A  e.  _V  ->  ran  {
<. A ,  B >. }  =  { B }
)
31, 2ax-mp 5 1  |-  ran  { <. A ,  B >. }  =  { B }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002   <.cop 4008   ran crn 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-dm 4864  df-rn 4865
This theorem is referenced by:  op2nda  5341  fpr  6087  en1  7643  fodomfi  7856  dcomex  8875  s1rn  12725  axlowdimlem13  24830  ex-rn  25735  ex-ima  25737  gidsn  25921  ginvsn  25922  rngosn  25977  zrdivrng  26005  ghomsn  30094  ptrest  31643  poimirlem3  31647
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