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Theorem rnsnop 5489
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 5487 . 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 1379    e. wcel 1767   _Vcvv 3113   {csn 4027   <.cop 4033   ran crn 5000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010
This theorem is referenced by:  op2nda  5493  fpr  6070  en1  7583  fodomfi  7800  dcomex  8828  s1rn  12577  axlowdimlem13  24030  ex-rn  24935  ex-ima  24937  gidsn  25123  ginvsn  25124  rngosn  25179  zrdivrng  25207  ghomsn  28779  ptrest  29901
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