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Theorem op2nda 5354
Description: Extract the second member of an ordered pair. (See op1sta 5351 to extract the first member, op2ndb 5353 for an alternate version, and op2nd 6356 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 5350 . . 3  |-  ran  { <. A ,  B >. }  =  { B }
32unieqi 4025 . 2  |-  U. ran  {
<. A ,  B >. }  =  U. { B }
4 cnvsn.2 . . 3  |-  B  e. 
_V
54unisn 4031 . 2  |-  U. { B }  =  B
63, 5eqtri 2456 1  |-  U. ran  {
<. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   <.cop 3817   U.cuni 4015   ran crn 4879
This theorem is referenced by:  elxp4  5357  elxp5  5358  op2nd  6356  fo2nd  6367  f2ndres  6369  ixpsnf1o  7102  xpassen  7202  xpdom2  7203  xpnnenOLD  12809
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889
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