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Theorem pjfval2 18866
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  =  ( proj1 `  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 4517 . . 3  |-  ( x  e.  ( LSubSp `  W
)  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
2 df-xp 5014 . . 3  |-  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) }
31, 2ineq12i 3694 . 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 2457 . . 3  |-  ( Base `  W )  =  (
Base `  W )
5 eqid 2457 . . 3  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
6 pjfval2.o . . 3  |-  ._|_  =  ( ocv `  W )
7 pjfval2.p . . 3  |-  P  =  ( proj1 `  W )
8 pjfval2.k . . 3  |-  K  =  ( proj `  W
)
94, 5, 6, 7, 8pjfval 18863 . 2  |-  K  =  ( ( x  e.  ( LSubSp `  W )  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) ) )
104, 5, 6, 7, 8pjdm 18864 . . . . . . 7  |-  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  ( x P ( 
._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) ) )
11 eleq1 2529 . . . . . . . . 9  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
12 fvex 5882 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
1312, 12elmap 7466 . . . . . . . . 9  |-  ( ( x P (  ._|_  `  x ) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) ) : (
Base `  W ) --> ( Base `  W )
)
1411, 13syl6rbb 262 . . . . . . . 8  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( (
x P (  ._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
)  <->  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
1514anbi2d 703 . . . . . . 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 257 . . . . . 6  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) ) ) )
1716pm5.32ri 638 . . . . 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 798 . . . . 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 3112 . . . . . . 7  |-  x  e. 
_V
2019biantrur 506 . . . . . 6  |-  ( y  e.  ( ( Base `  W )  ^m  ( Base `  W ) )  <-> 
( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) )
2120anbi2i 694 . . . . 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 271 . . . 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 4521 . . 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 4517 . . 3  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
25 inopab 5143 . . 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 2496 . 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 2496 1  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470   {copab 4514    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   -->wf 5590   ` cfv 5594  (class class class)co 6296    ^m cmap 7438   Basecbs 14643   proj1cpj1 16781   LSubSpclss 17704   ocvcocv 18817   projcpj 18857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-pj 18860
This theorem is referenced by:  pjval  18867  pjff  18869
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