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Theorem op2ndg 6396
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 4013 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5767 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2451 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 4014 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5767 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 21 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2457 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2968 . . 3  |-  x  e. 
_V
9 vex 2968 . . 3  |-  y  e. 
_V
108, 9op2nd 6392 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 3024 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1654    e. wcel 1728   <.cop 3846   ` cfv 5489   2ndc2nd 6384
This theorem is referenced by:  ot2ndg  6398  ot3rdg  6399  2ndconst  6472  curry1  6474  xpmapenlem  7310  axdc4lem  8373  pinq  8842  addpipq  8852  mulpipq  8855  ordpipq  8857  swrdval  11802  ruclem1  12868  eucalg  13116  qnumdenbi  13174  comffval  13963  oppccofval  13980  funcf2  14103  cofuval2  14122  resfval2  14128  resf2nd  14130  funcres  14131  isnat  14182  fucco  14197  homacd  14234  setcco  14276  catcco  14294  xpcco  14318  xpchom2  14321  xpcco2  14322  evlf2  14353  curfval  14358  curf1cl  14363  uncf1  14371  uncf2  14372  hof2fval  14390  yonedalem21  14408  yonedalem22  14413  imasdsf1olem  18441  ovolicc1  19450  ioombl1lem3  19492  ioombl1lem4  19493  nbgraop  21474  vcoprne  22096  brcgr  25874  fvtransport  26001  dvhopvadd  32065  dvhopvsca  32074  dvhopaddN  32086  dvhopspN  32087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5453  df-fun 5491  df-fv 5497  df-2nd 6386
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