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Theorem op2nda 5157
Description: Extract the second member of an ordered pair. (See op1sta 5154 to extract the first member, op2ndb 5156 for an alternate version, and op2nd 6129 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
cnvsn.1  |-  A  e. 
_V
cnvsn.2  |-  B  e. 
_V
Assertion
Ref Expression
op2nda  |-  U. ran  {
<. A ,  B >. }  =  B

Proof of Theorem op2nda
StepHypRef Expression
1 cnvsn.1 . . . 4  |-  A  e. 
_V
21rnsnop 5153 . . 3  |-  ran  { <. A ,  B >. }  =  { B }
32unieqi 3837 . 2  |-  U. ran  {
<. A ,  B >. }  =  U. { B }
4 cnvsn.2 . . 3  |-  B  e. 
_V
54unisn 3843 . 2  |-  U. { B }  =  B
63, 5eqtri 2303 1  |-  U. ran  {
<. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   <.cop 3643   U.cuni 3827   ran crn 4690
This theorem is referenced by:  elxp4  5160  elxp5  5161  op2nd  6129  fo2nd  6140  f2ndres  6142  ixpsnf1o  6856  xpassen  6956  xpdom2  6957  xpnnenOLD  12488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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