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Theorem op2ndg 6794
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 4213 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5868 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2469 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 4214 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5868 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 22 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2489 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 3116 . . 3  |-  x  e. 
_V
9 vex 3116 . . 3  |-  y  e. 
_V
108, 9op2nd 6790 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 3175 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4033   ` cfv 5586   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-2nd 6782
This theorem is referenced by:  ot2ndg  6796  ot3rdg  6797  2ndconst  6869  mpt2sn  6871  curry1  6872  xpmapenlem  7681  axdc4lem  8831  pinq  9301  addpipq  9311  mulpipq  9314  ordpipq  9316  swrdval  12603  ruclem1  13821  eucalg  14071  qnumdenbi  14132  comffval  14951  oppccofval  14968  funcf2  15091  cofuval2  15110  resfval2  15116  resf2nd  15118  funcres  15119  isnat  15170  fucco  15185  homacd  15222  setcco  15264  catcco  15282  xpcco  15306  xpchom2  15309  xpcco2  15310  evlf2  15341  curfval  15346  curf1cl  15351  uncf1  15359  uncf2  15360  hof2fval  15378  yonedalem21  15396  yonedalem22  15401  mvmulfval  18811  imasdsf1olem  20611  ovolicc1  21662  ioombl1lem3  21705  ioombl1lem4  21706  brcgr  23879  edgopval  24016  nbgraop  24099  nbgraopALT  24100  vcoprne  25148  fvtransport  29259  gsizopval  31860  lmod1zr  32175  bj-finsumval0  33735  dvhopvadd  35890  dvhopvsca  35899  dvhopaddN  35911  dvhopspN  35912
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