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Theorem prj3 25183
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 2804 . . . 4  |-  y  e. 
_V
21elima 5033 . . 3  |-  ( y  e.  ( 2nd " R
)  <->  E. z  e.  R  z 2nd y )
3 df-rel 4712 . . . . . . . . 9  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
4 ssel 3187 . . . . . . . . . . 11  |-  ( R 
C_  ( _V  X.  _V )  ->  ( z  e.  R  ->  z  e.  ( _V  X.  _V ) ) )
5 elxp6 6167 . . . . . . . . . . . 12  |-  ( z  e.  ( _V  X.  _V )  <->  ( z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >.  /\  (
( 1st `  z
)  e.  _V  /\  ( 2nd `  z )  e.  _V ) ) )
6 df-br 4040 . . . . . . . . . . . . . . 15  |-  ( z 2nd y  <->  <. z ,  y >.  e.  2nd )
7 fo2nd 6156 . . . . . . . . . . . . . . . . . 18  |-  2nd : _V -onto-> _V
8 fofn 5469 . . . . . . . . . . . . . . . . . 18  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
97, 8ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  2nd  Fn  _V
10 vex 2804 . . . . . . . . . . . . . . . . 17  |-  z  e. 
_V
11 fnopfvb 5580 . . . . . . . . . . . . . . . . 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 2356 . . . . . . . . . . . . . . . . . 18  |-  ( z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  ->  (
z  e.  R  <->  <. ( 1st `  z ) ,  ( 2nd `  z )
>.  e.  R ) )
14 opeq2 3813 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  z )  =  y  ->  <. ( 1st `  z ) ,  ( 2nd `  z
) >.  =  <. ( 1st `  z ) ,  y >. )
1514eleq1d 2362 . . . . . . . . . . . . . . . . . . 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 5555 . . . . . . . 8  |-  ( 1st `  z )  e.  _V
29 opeq1 3812 . . . . . . . . 9  |-  ( x  =  ( 1st `  z
)  ->  <. x ,  y >.  =  <. ( 1st `  z ) ,  y >. )
3029eleq1d 2362 . . . . . . . 8  |-  ( x  =  ( 1st `  z
)  ->  ( <. x ,  y >.  e.  R  <->  <.
( 1st `  z
) ,  y >.  e.  R ) )
3128, 30spcev 2888 . . . . . . 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 2674 . . . . 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 2804 . . . . . . . . 9  |-  x  e. 
_V
3635, 1op2nd 6145 . . . . . . . 8  |-  ( 2nd `  <. x ,  y
>. )  =  y
37 opex 4253 . . . . . . . . 9  |-  <. x ,  y >.  e.  _V
38 fnopfvb 5580 . . . . . . . . 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 4040 . . . . . . 7  |-  ( <.
x ,  y >. 2nd y  <->  <. <. x ,  y
>. ,  y >.  e. 
2nd )
4240, 41mpbir 200 . . . . . 6  |-  <. x ,  y >. 2nd y
43 breq1 4042 . . . . . . 7  |-  ( z  =  <. x ,  y
>.  ->  ( z 2nd y  <->  <. x ,  y
>. 2nd y ) )
4443rspcev 2897 . . . . . 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 1624 . . . 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 2411 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    C_ wss 3165   <.cop 3656   class class class wbr 4039    X. cxp 4703   "cima 4708   Rel wrel 4710    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  prjrn  25186
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139
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