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Theorem op2nda 4881
Description: Extract the second member of an ordered pair. (See op1sta 4878 to extract the first member and op2ndb 4880 for an alternate 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 4877 . . 3  |-  ran  { <. A ,  B >. }  =  { B }
32unieqi 3646 . 2  |-  U. ran  {
<. A ,  B >. }  =  U. { B }
4 cnvsn.2 . . 3  |-  B  e. 
_V
54unisn 3652 . 2  |-  U. { B }  =  B
63, 5eqtri 2105 1  |-  U. ran  {
<. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1287    e. wcel 1436   _Vcvv 2615   {csn 3431   <.cop 3434   U.cuni 3636   ran crn 4412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-xp 4417  df-rel 4418  df-cnv 4419  df-dm 4421  df-rn 4422
This theorem is referenced by:  elxp4  4884  elxp5  4885  op2nd  5875  fo2nd  5886  f2ndres  5888  xpassen  6498  xpdom2  6499
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