MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2ndg Structured version   Unicode version

Theorem op2ndg 6695
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 4162 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5798 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2454 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 4163 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5798 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 22 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2474 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 3075 . . 3  |-  x  e. 
_V
9 vex 3075 . . 3  |-  y  e. 
_V
108, 9op2nd 6691 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 3134 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   <.cop 3986   ` cfv 5521   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fv 5529  df-2nd 6683
This theorem is referenced by:  ot2ndg  6697  ot3rdg  6698  2ndconst  6767  curry1  6769  xpmapenlem  7583  axdc4lem  8730  pinq  9202  addpipq  9212  mulpipq  9215  ordpipq  9217  swrdval  12426  ruclem1  13626  eucalg  13875  qnumdenbi  13935  comffval  14752  oppccofval  14769  funcf2  14892  cofuval2  14911  resfval2  14917  resf2nd  14919  funcres  14920  isnat  14971  fucco  14986  homacd  15023  setcco  15065  catcco  15083  xpcco  15107  xpchom2  15110  xpcco2  15111  evlf2  15142  curfval  15147  curf1cl  15152  uncf1  15160  uncf2  15161  hof2fval  15179  yonedalem21  15197  yonedalem22  15202  mvmulfval  18475  imasdsf1olem  20075  ovolicc1  21126  ioombl1lem3  21169  ioombl1lem4  21170  brcgr  23293  nbgraop  23482  vcoprne  24104  fvtransport  28202  mpt2sn  30866  lmod1zr  31149  bj-finsumval0  32902  dvhopvadd  35057  dvhopvsca  35066  dvhopaddN  35078  dvhopspN  35079
  Copyright terms: Public domain W3C validator