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Theorem op2ndd 6047
Description: Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
op1st.1  |-  A  e. 
_V
op1st.2  |-  B  e. 
_V
Assertion
Ref Expression
op2ndd  |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C
)  =  B )

Proof of Theorem op2ndd
StepHypRef Expression
1 fveq2 5421 . 2  |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C
)  =  ( 2nd `  <. A ,  B >. ) )
2 op1st.1 . . 3  |-  A  e. 
_V
3 op1st.2 . . 3  |-  B  e. 
_V
42, 3op2nd 6045 . 2  |-  ( 2nd `  <. A ,  B >. )  =  B
51, 4syl6eq 2188 1  |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   _Vcvv 2686   <.cop 3530   ` cfv 5123   2ndc2nd 6037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039
This theorem is referenced by:  xp2nd  6064  sbcopeq1a  6085  csbopeq1a  6086  eloprabi  6094  mpomptsx  6095  dmmpossx  6097  fmpox  6098  fmpoco  6113  df2nd2  6117  xporderlem  6128  xpf1o  6738  frecuzrdgtcl  10197  frecuzrdgfunlem  10204  fisumcom2  11219  txbas  12441  cnmpt2nd  12472  txhmeo  12502
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