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Theorem rnsnop 5431
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 5429 . 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 1370    e. wcel 1758   _Vcvv 3078   {csn 3988   <.cop 3994   ran crn 4952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-dm 4961  df-rn 4962
This theorem is referenced by:  op2nda  5435  fpr  6002  en1  7489  fodomfi  7704  dcomex  8730  axlowdimlem13  23372  ex-rn  23819  ex-ima  23821  gidsn  24007  ginvsn  24008  rngosn  24063  zrdivrng  24091  ghomsn  27471  ptrest  28593
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