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Theorem op2ndg 6149
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 3812 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5545 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2304 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3813 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5545 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 19 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2310 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2804 . . 3  |-  x  e. 
_V
9 vex 2804 . . 3  |-  y  e. 
_V
108, 9op2nd 6145 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 2860 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   ` cfv 5271   2ndc2nd 6137
This theorem is referenced by:  ot2ndg  6151  ot3rdg  6152  2ndconst  6224  curry1  6226  xpmapenlem  7044  axdc4lem  8097  pinq  8567  addpipq  8577  mulpipq  8580  ordpipq  8582  swrdval  11466  ruclem1  12525  eucalg  12773  qnumdenbi  12831  comffval  13618  oppccofval  13635  funcf2  13758  cofuval2  13777  resfval2  13783  resf2nd  13785  funcres  13786  isnat  13837  fucco  13852  homacd  13889  setcco  13931  catcco  13949  xpcco  13973  xpchom2  13976  xpcco2  13977  evlf2  14008  curfval  14013  curf1cl  14018  uncf1  14026  uncf2  14027  hof2fval  14045  yonedalem21  14063  yonedalem22  14068  imasdsf1olem  17953  ovolicc1  18891  ioombl1lem3  18933  ioombl1lem4  18934  vcoprne  21151  brcgr  24600  fvtransport  24727  codidmor  26053  grphidmor  26055  grphidmor2  26056  nbgraop  28274  nbusgra  28277  dvhopvadd  31905  dvhopvsca  31914  dvhopaddN  31926  dvhopspN  31927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fv 5279  df-2nd 6139
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