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Theorem op2ndg 6133
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 3796 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5529 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2291 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3797 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5529 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 19 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2297 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2791 . . 3  |-  x  e. 
_V
9 vex 2791 . . 3  |-  y  e. 
_V
108, 9op2nd 6129 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 2847 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   ` cfv 5255   2ndc2nd 6121
This theorem is referenced by:  ot2ndg  6135  ot3rdg  6136  2ndconst  6208  curry1  6210  xpmapenlem  7028  axdc4lem  8081  pinq  8551  addpipq  8561  mulpipq  8564  ordpipq  8566  swrdval  11450  ruclem1  12509  eucalg  12757  qnumdenbi  12815  comffval  13602  oppccofval  13619  funcf2  13742  cofuval2  13761  resfval2  13767  resf2nd  13769  funcres  13770  isnat  13821  fucco  13836  homacd  13873  setcco  13915  catcco  13933  xpcco  13957  xpchom2  13960  xpcco2  13961  evlf2  13992  curfval  13997  curf1cl  14002  uncf1  14010  uncf2  14011  hof2fval  14029  yonedalem21  14047  yonedalem22  14052  imasdsf1olem  17937  ovolicc1  18875  ioombl1lem3  18917  ioombl1lem4  18918  vcoprne  21135  brcgr  23939  fvtransport  24066  codidmor  25362  grphidmor  25364  grphidmor2  25365  dvhopvadd  30656  dvhopvsca  30665  dvhopaddN  30677  dvhopspN  30678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fv 5263  df-2nd 6123
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