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Theorem op2ndg 6015
Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
Assertion
Ref Expression
op2ndg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )

Proof of Theorem op2ndg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3673 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5391 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2124 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3674 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5391 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 19 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2130 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2661 . . 3  |-  x  e. 
_V
9 vex 2661 . . 3  |-  y  e. 
_V
108, 9op2nd 6011 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 2722 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   <.cop 3498   ` cfv 5091   2ndc2nd 6003
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fv 5099  df-2nd 6005
This theorem is referenced by:  ot2ndg  6017  ot3rdgg  6018  2ndconst  6085  xpmapenlem  6709  2ndinl  6926  2ndinr  6928  mulpipq  7144  suplocexprlem2b  7486  aprcl  8371  frec2uzrdg  10133  frecuzrdgsuc  10138  eucalglt  11645  eucalg  11647  qredeu  11685  sqpweven  11759  2sqpwodd  11760  qnumdenbi  11776  upxp  12347  uptx  12349  txmetcnp  12593
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