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

Theorem op2ndg 6352
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 3976 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5724 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2443 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3977 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5724 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 20 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2449 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2951 . . 3  |-  x  e. 
_V
9 vex 2951 . . 3  |-  y  e. 
_V
108, 9op2nd 6348 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 3007 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3809   ` cfv 5446   2ndc2nd 6340
This theorem is referenced by:  ot2ndg  6354  ot3rdg  6355  2ndconst  6428  curry1  6430  xpmapenlem  7266  axdc4lem  8327  pinq  8796  addpipq  8806  mulpipq  8809  ordpipq  8811  swrdval  11756  ruclem1  12822  eucalg  13070  qnumdenbi  13128  comffval  13917  oppccofval  13934  funcf2  14057  cofuval2  14076  resfval2  14082  resf2nd  14084  funcres  14085  isnat  14136  fucco  14151  homacd  14188  setcco  14230  catcco  14248  xpcco  14272  xpchom2  14275  xpcco2  14276  evlf2  14307  curfval  14312  curf1cl  14317  uncf1  14325  uncf2  14326  hof2fval  14344  yonedalem21  14362  yonedalem22  14367  imasdsf1olem  18395  ovolicc1  19404  ioombl1lem3  19446  ioombl1lem4  19447  nbgraop  21428  vcoprne  22050  brcgr  25831  fvtransport  25958  dvhopvadd  31828  dvhopvsca  31837  dvhopaddN  31849  dvhopspN  31850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-2nd 6342
  Copyright terms: Public domain W3C validator