ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  op2ndg Unicode version

Theorem op2ndg 5904
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 3617 . . . 4  |-  ( x  =  A  ->  <. x ,  y >.  =  <. A ,  y >. )
21fveq2d 5293 . . 3  |-  ( x  =  A  ->  ( 2nd `  <. x ,  y
>. )  =  ( 2nd `  <. A ,  y
>. ) )
32eqeq1d 2096 . 2  |-  ( x  =  A  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  ( 2nd `  <. A ,  y >. )  =  y ) )
4 opeq2 3618 . . . 4  |-  ( y  =  B  ->  <. A , 
y >.  =  <. A ,  B >. )
54fveq2d 5293 . . 3  |-  ( y  =  B  ->  ( 2nd `  <. A ,  y
>. )  =  ( 2nd `  <. A ,  B >. ) )
6 id 19 . . 3  |-  ( y  =  B  ->  y  =  B )
75, 6eqeq12d 2102 . 2  |-  ( y  =  B  ->  (
( 2nd `  <. A ,  y >. )  =  y  <->  ( 2nd `  <. A ,  B >. )  =  B ) )
8 vex 2622 . . 3  |-  x  e. 
_V
9 vex 2622 . . 3  |-  y  e. 
_V
108, 9op2nd 5900 . 2  |-  ( 2nd `  <. x ,  y
>. )  =  y
113, 7, 10vtocl2g 2683 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 2nd `  <. A ,  B >. )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   <.cop 3444   ` cfv 5002   2ndc2nd 5892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fv 5010  df-2nd 5894
This theorem is referenced by:  ot2ndg  5906  ot3rdgg  5907  2ndconst  5969  xpmapenlem  6545  2ndinl  6745  2ndinr  6747  mulpipq  6910  frec2uzrdg  9781  frecuzrdgsuc  9786  eucalglt  11132  eucialg  11134  qredeu  11172  sqpweven  11246  2sqpwodd  11247  qnumdenbi  11263
  Copyright terms: Public domain W3C validator