MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pjfval2 Unicode version

Theorem pjfval2 16609
Description: Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval2.o  |-  ._|_  =  ( ocv `  W )
pjfval2.p  |-  P  =  ( proj 1 `  W )
pjfval2.k  |-  K  =  ( proj `  W
)
Assertion
Ref Expression
pjfval2  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Distinct variable groups:    x, K    x, 
._|_    x, P    x, W

Proof of Theorem pjfval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4079 . . 3  |-  ( x  e.  ( LSubSp `  W
)  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
2 df-xp 4695 . . 3  |-  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) }
31, 2ineq12i 3368 . 2  |-  ( ( x  e.  ( LSubSp `  W )  |->  ( x P (  ._|_  `  x
) ) )  i^i  ( _V  X.  (
( Base `  W )  ^m  ( Base `  W
) ) ) )  =  ( { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x ) ) ) }  i^i  {
<. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) } )
4 eqid 2283 . . 3  |-  ( Base `  W )  =  (
Base `  W )
5 eqid 2283 . . 3  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 pjfval2.o . . 3  |-  ._|_  =  ( ocv `  W )
7 pjfval2.p . . 3  |-  P  =  ( proj 1 `  W )
8 pjfval2.k . . 3  |-  K  =  ( proj `  W
)
94, 5, 6, 7, 8pjfval 16606 . 2  |-  K  =  ( ( x  e.  ( LSubSp `  W )  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) ) )
104, 5, 6, 7, 8pjdm 16607 . . . . . . 7  |-  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  ( x P ( 
._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) ) )
11 eleq1 2343 . . . . . . . . 9  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
12 fvex 5539 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
1312, 12elmap 6796 . . . . . . . . 9  |-  ( ( x P (  ._|_  `  x ) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) ) : (
Base `  W ) --> ( Base `  W )
)
1411, 13syl6rbb 253 . . . . . . . 8  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( (
x P (  ._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
)  <->  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
1514anbi2d 684 . . . . . . 7  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( (
x  e.  ( LSubSp `  W )  /\  (
x P (  ._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) )  <->  ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) ) ) )
1610, 15syl5bb 248 . . . . . 6  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) ) ) )
1716pm5.32ri 619 . . . . 5  |-  ( ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) )  <-> 
( ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) )  /\  y  =  ( x P (  ._|_  `  x
) ) ) )
18 an32 773 . . . . 5  |-  ( ( ( x  e.  (
LSubSp `  W )  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) )  /\  y  =  ( x P (  ._|_  `  x ) ) )  <->  ( (
x  e.  ( LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x ) ) )  /\  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
19 vex 2791 . . . . . . 7  |-  x  e. 
_V
2019biantrur 492 . . . . . 6  |-  ( y  e.  ( ( Base `  W )  ^m  ( Base `  W ) )  <-> 
( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) )
2120anbi2i 675 . . . . 5  |-  ( ( ( x  e.  (
LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x
) ) )  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) )  <->  ( (
x  e.  ( LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x ) ) )  /\  (
x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) ) )
2217, 18, 213bitri 262 . . . 4  |-  ( ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) )  <-> 
( ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) )  /\  ( x  e. 
_V  /\  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) ) )
2322opabbii 4083 . . 3  |-  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }  =  { <. x ,  y >.  |  ( ( x  e.  (
LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x
) ) )  /\  ( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) ) }
24 df-mpt 4079 . . 3  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
25 inopab 4816 . . 3  |-  ( {
<. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }  i^i  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) } )  =  { <. x ,  y >.  |  ( ( x  e.  (
LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x
) ) )  /\  ( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) ) }
2623, 24, 253eqtr4i 2313 . 2  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  ( { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  ( x P (  ._|_  `  x ) ) ) }  i^i  {
<. x ,  y >.  |  ( x  e. 
_V  /\  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) } )
273, 9, 263eqtr4i 2313 1  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   {copab 4076    e. cmpt 4077    X. cxp 4687   dom cdm 4689   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Basecbs 13148   proj 1cpj1 14946   LSubSpclss 15689   ocvcocv 16560   projcpj 16600
This theorem is referenced by:  pjval  16610  pjff  16612
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-pw 3627  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-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-pj 16603
  Copyright terms: Public domain W3C validator