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Theorem pjfval2 16936
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 4268 . . 3  |-  ( x  e.  ( LSubSp `  W
)  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
2 df-xp 4884 . . 3  |-  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) }
31, 2ineq12i 3540 . 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 2436 . . 3  |-  ( Base `  W )  =  (
Base `  W )
5 eqid 2436 . . 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 16933 . 2  |-  K  =  ( ( x  e.  ( LSubSp `  W )  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) ) )
104, 5, 6, 7, 8pjdm 16934 . . . . . . 7  |-  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  ( x P ( 
._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) ) )
11 eleq1 2496 . . . . . . . . 9  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
12 fvex 5742 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
1312, 12elmap 7042 . . . . . . . . 9  |-  ( ( x P (  ._|_  `  x ) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) ) : (
Base `  W ) --> ( Base `  W )
)
1411, 13syl6rbb 254 . . . . . . . 8  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( (
x P (  ._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
)  <->  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
1514anbi2d 685 . . . . . . 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 249 . . . . . 6  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) ) ) )
1716pm5.32ri 620 . . . . 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 774 . . . . 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 2959 . . . . . . 7  |-  x  e. 
_V
2019biantrur 493 . . . . . 6  |-  ( y  e.  ( ( Base `  W )  ^m  ( Base `  W ) )  <-> 
( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) )
2120anbi2i 676 . . . . 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 263 . . . 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 4272 . . 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 4268 . . 3  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
25 inopab 5005 . . 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 2466 . 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 2466 1  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   {copab 4265    e. cmpt 4266    X. cxp 4876   dom cdm 4878   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^m cmap 7018   Basecbs 13469   proj 1cpj1 15269   LSubSpclss 16008   ocvcocv 16887   projcpj 16927
This theorem is referenced by:  pjval  16937  pjff  16939
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-pj 16930
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