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Theorem op2ndd 5807
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 5209 . 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 5805 . 2  |-  ( 2nd `  <. A ,  B >. )  =  B
51, 4syl6eq 2130 1  |-  ( C  =  <. A ,  B >.  ->  ( 2nd `  C
)  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   _Vcvv 2602   <.cop 3409   ` cfv 4932   2ndc2nd 5797
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-2nd 5799
This theorem is referenced by:  xp2nd  5824  sbcopeq1a  5844  csbopeq1a  5845  eloprabi  5853  mpt2mptsx  5854  dmmpt2ssx  5856  fmpt2x  5857  fmpt2co  5868  df2nd2  5872  xporderlem  5883  xpf1o  6385  frecuzrdgtcl  9494  frecuzrdgfunlem  9501
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