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Theorem op2nda 5173
Description: Extract the second member of an ordered pair. (See op1sta 5170 to extract the first member, op2ndb 5172 for an alternate version, and op2nd 6145 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 5169 . . 3  |-  ran  { <. A ,  B >. }  =  { B }
32unieqi 3853 . 2  |-  U. ran  {
<. A ,  B >. }  =  U. { B }
4 cnvsn.2 . . 3  |-  B  e. 
_V
54unisn 3859 . 2  |-  U. { B }  =  B
63, 5eqtri 2316 1  |-  U. ran  {
<. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   U.cuni 3843   ran crn 4706
This theorem is referenced by:  elxp4  5176  elxp5  5177  op2nd  6145  fo2nd  6156  f2ndres  6158  ixpsnf1o  6872  xpassen  6972  xpdom2  6973  xpnnenOLD  12504
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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