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Theorem op2ndg 6299
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 3926 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5672 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2395 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3927 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5672 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 20 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2401 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2902 . . 3  |-  x  e. 
_V
9 vex 2902 . . 3  |-  y  e. 
_V
108, 9op2nd 6295 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 2958 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3760   ` cfv 5394   2ndc2nd 6287
This theorem is referenced by:  ot2ndg  6301  ot3rdg  6302  2ndconst  6375  curry1  6377  xpmapenlem  7210  axdc4lem  8268  pinq  8737  addpipq  8747  mulpipq  8750  ordpipq  8752  swrdval  11691  ruclem1  12757  eucalg  13005  qnumdenbi  13063  comffval  13852  oppccofval  13869  funcf2  13992  cofuval2  14011  resfval2  14017  resf2nd  14019  funcres  14020  isnat  14071  fucco  14086  homacd  14123  setcco  14165  catcco  14183  xpcco  14207  xpchom2  14210  xpcco2  14211  evlf2  14242  curfval  14247  curf1cl  14252  uncf1  14260  uncf2  14261  hof2fval  14279  yonedalem21  14297  yonedalem22  14302  imasdsf1olem  18311  ovolicc1  19279  ioombl1lem3  19321  ioombl1lem4  19322  nbgraop  21302  nbusgra  21306  vcoprne  21906  brcgr  25553  fvtransport  25680  dvhopvadd  31208  dvhopvsca  31217  dvhopaddN  31229  dvhopspN  31230
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-iota 5358  df-fun 5396  df-fv 5402  df-2nd 6289
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