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Theorem op2nda 5115
Description: Extract the second member of an ordered pair. (See op1sta 5112 to extract the first member and op2ndb 5114 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 5111 . . 3  |-  ran  { <. A ,  B >. }  =  { B }
32unieqi 3821 . 2  |-  U. ran  {
<. A ,  B >. }  =  U. { B }
4 cnvsn.2 . . 3  |-  B  e. 
_V
54unisn 3827 . 2  |-  U. { B }  =  B
63, 5eqtri 2198 1  |-  U. ran  {
<. A ,  B >. }  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148   _Vcvv 2739   {csn 3594   <.cop 3597   U.cuni 3811   ran crn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636  df-dm 4638  df-rn 4639
This theorem is referenced by:  elxp4  5118  elxp5  5119  op2nd  6150  fo2nd  6161  f2ndres  6163  ixpsnf1o  6738  xpassen  6832  xpdom2  6833
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