Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prj3 Unicode version

Theorem prj3 25080
Description: Projection of the second elements of the pairs of a relation 
R. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
prj3  |-  ( Rel 
R  ->  ( 2nd " R )  =  {
y  |  E. x <. x ,  y >.  e.  R } )
Distinct variable group:    x, R, y

Proof of Theorem prj3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . 4  |-  y  e. 
_V
21elima 5017 . . 3  |-  ( y  e.  ( 2nd " R
)  <->  E. z  e.  R  z 2nd y )
3 df-rel 4696 . . . . . . . . 9  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
4 ssel 3174 . . . . . . . . . . 11  |-  ( R 
C_  ( _V  X.  _V )  ->  ( z  e.  R  ->  z  e.  ( _V  X.  _V ) ) )
5 elxp6 6151 . . . . . . . . . . . 12  |-  ( z  e.  ( _V  X.  _V )  <->  ( z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  _V  /\  ( 2nd `  z )  e.  _V ) ) )
6 df-br 4024 . . . . . . . . . . . . . . 15  |-  ( z 2nd y  <->  <. z ,  y >.  e.  2nd )
7 fo2nd 6140 . . . . . . . . . . . . . . . . . 18  |-  2nd : _V -onto-> _V
8 fofn 5453 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
97, 8ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  2nd  Fn  _V
10 vex 2791 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
11 fnopfvb 5564 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd  Fn  _V  /\  z  e.  _V )  ->  ( ( 2nd `  z
)  =  y  <->  <. z ,  y >.  e.  2nd ) )
129, 10, 11mp2an 653 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  z )  =  y  <->  <. z ,  y >.  e.  2nd )
13 eleq1 2343 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
z  e.  R  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  R ) )
14 opeq2 3797 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  z )  =  y  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  =  <. ( 1st `  z ) ,  y >. )
1514eleq1d 2349 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  z )  =  y  ->  ( <. ( 1st `  z
) ,  ( 2nd `  z ) >.  e.  R  <->  <.
( 1st `  z
) ,  y >.  e.  R ) )
1615biimpcd 215 . . . . . . . . . . . . . . . . . 18  |-  ( <.
( 1st `  z
) ,  ( 2nd `  z ) >.  e.  R  ->  ( ( 2nd `  z
)  =  y  ->  <. ( 1st `  z
) ,  y >.  e.  R ) )
1713, 16syl6bi 219 . . . . . . . . . . . . . . . . 17  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
z  e.  R  -> 
( ( 2nd `  z
)  =  y  ->  <. ( 1st `  z
) ,  y >.  e.  R ) ) )
1817com3r 73 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  z )  =  y  ->  (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  ->  ( z  e.  R  ->  <. ( 1st `  z ) ,  y >.  e.  R
) ) )
1912, 18sylbir 204 . . . . . . . . . . . . . . 15  |-  ( <.
z ,  y >.  e.  2nd  ->  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
z  e.  R  ->  <. ( 1st `  z
) ,  y >.  e.  R ) ) )
206, 19sylbi 187 . . . . . . . . . . . . . 14  |-  ( z 2nd y  ->  (
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  ->  ( z  e.  R  ->  <. ( 1st `  z ) ,  y >.  e.  R
) ) )
2120com3l 75 . . . . . . . . . . . . 13  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
z  e.  R  -> 
( z 2nd y  -> 
<. ( 1st `  z
) ,  y >.  e.  R ) ) )
2221adantr 451 . . . . . . . . . . . 12  |-  ( ( z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  /\  ( ( 1st `  z )  e. 
_V  /\  ( 2nd `  z )  e.  _V ) )  ->  (
z  e.  R  -> 
( z 2nd y  -> 
<. ( 1st `  z
) ,  y >.  e.  R ) ) )
235, 22sylbi 187 . . . . . . . . . . 11  |-  ( z  e.  ( _V  X.  _V )  ->  ( z  e.  R  ->  (
z 2nd y  ->  <. ( 1st `  z
) ,  y >.  e.  R ) ) )
244, 23syli 33 . . . . . . . . . 10  |-  ( R 
C_  ( _V  X.  _V )  ->  ( z  e.  R  ->  (
z 2nd y  ->  <. ( 1st `  z
) ,  y >.  e.  R ) ) )
2524com23 72 . . . . . . . . 9  |-  ( R 
C_  ( _V  X.  _V )  ->  ( z 2nd y  ->  (
z  e.  R  ->  <. ( 1st `  z
) ,  y >.  e.  R ) ) )
263, 25sylbi 187 . . . . . . . 8  |-  ( Rel 
R  ->  ( z 2nd y  ->  ( z  e.  R  ->  <. ( 1st `  z ) ,  y >.  e.  R
) ) )
2726com13 74 . . . . . . 7  |-  ( z  e.  R  ->  (
z 2nd y  -> 
( Rel  R  ->  <.
( 1st `  z
) ,  y >.  e.  R ) ) )
28 fvex 5539 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
29 opeq1 3796 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  <. x ,  y >.  =  <. ( 1st `  z ) ,  y >. )
3029eleq1d 2349 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( <. x ,  y >.  e.  R  <->  <.
( 1st `  z
) ,  y >.  e.  R ) )
3128, 30spcev 2875 . . . . . . 7  |-  ( <.
( 1st `  z
) ,  y >.  e.  R  ->  E. x <. x ,  y >.  e.  R )
3227, 31syl8 65 . . . . . 6  |-  ( z  e.  R  ->  (
z 2nd y  -> 
( Rel  R  ->  E. x <. x ,  y
>.  e.  R ) ) )
3332rexlimiv 2661 . . . . 5  |-  ( E. z  e.  R  z 2nd y  ->  ( Rel  R  ->  E. x <. x ,  y >.  e.  R ) )
3433com12 27 . . . 4  |-  ( Rel 
R  ->  ( E. z  e.  R  z 2nd y  ->  E. x <. x ,  y >.  e.  R ) )
35 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
3635, 1op2nd 6129 . . . . . . . 8  |-  ( 2nd `  <. x ,  y
>. )  =  y
37 opex 4237 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
38 fnopfvb 5564 . . . . . . . . 9  |-  ( ( 2nd  Fn  _V  /\  <.
x ,  y >.  e.  _V )  ->  (
( 2nd `  <. x ,  y >. )  =  y  <->  <. <. x ,  y
>. ,  y >.  e. 
2nd ) )
399, 37, 38mp2an 653 . . . . . . . 8  |-  ( ( 2nd `  <. x ,  y >. )  =  y  <->  <. <. x ,  y
>. ,  y >.  e. 
2nd )
4036, 39mpbi 199 . . . . . . 7  |-  <. <. x ,  y >. ,  y
>.  e.  2nd
41 df-br 4024 . . . . . . 7  |-  ( <.
x ,  y >. 2nd y  <->  <. <. x ,  y
>. ,  y >.  e. 
2nd )
4240, 41mpbir 200 . . . . . 6  |-  <. x ,  y >. 2nd y
43 breq1 4026 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z 2nd y  <->  <. x ,  y
>. 2nd y ) )
4443rspcev 2884 . . . . . 6  |-  ( (
<. x ,  y >.  e.  R  /\  <. x ,  y >. 2nd y
)  ->  E. z  e.  R  z 2nd y )
4542, 44mpan2 652 . . . . 5  |-  ( <.
x ,  y >.  e.  R  ->  E. z  e.  R  z 2nd y )
4645exlimiv 1666 . . . 4  |-  ( E. x <. x ,  y
>.  e.  R  ->  E. z  e.  R  z 2nd y )
4734, 46impbid1 194 . . 3  |-  ( Rel 
R  ->  ( E. z  e.  R  z 2nd y  <->  E. x <. x ,  y >.  e.  R
) )
482, 47syl5bb 248 . 2  |-  ( Rel 
R  ->  ( y  e.  ( 2nd " R
)  <->  E. x <. x ,  y >.  e.  R
) )
4948abbi2dv 2398 1  |-  ( Rel 
R  ->  ( 2nd " R )  =  {
y  |  E. x <. x ,  y >.  e.  R } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    C_ wss 3152   <.cop 3643   class class class wbr 4023    X. cxp 4687   "cima 4692   Rel wrel 4694    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  prjrn  25083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123
  Copyright terms: Public domain W3C validator