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Theorem pjfval2 19207
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 4420 . . 3  |-  ( x  e.  ( LSubSp `  W
)  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
2 df-xp 4795 . . 3  |-  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) }
31, 2ineq12i 3598 . 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 2422 . . 3  |-  ( Base `  W )  =  (
Base `  W )
5 eqid 2422 . . 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 19204 . 2  |-  K  =  ( ( x  e.  ( LSubSp `  W )  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) ) )
104, 5, 6, 7, 8pjdm 19205 . . . . . . 7  |-  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  ( x P ( 
._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) ) )
11 eleq1 2488 . . . . . . . . 9  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
12 fvex 5828 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
1312, 12elmap 7448 . . . . . . . . 9  |-  ( ( x P (  ._|_  `  x ) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) ) : (
Base `  W ) --> ( Base `  W )
)
1411, 13syl6rbb 265 . . . . . . . 8  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( (
x P (  ._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
)  <->  y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
1514anbi2d 708 . . . . . . 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 260 . . . . . 6  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  y  e.  (
( Base `  W )  ^m  ( Base `  W
) ) ) ) )
1716pm5.32ri 642 . . . . 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 805 . . . . 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 3019 . . . . . . 7  |-  x  e. 
_V
2019biantrur 508 . . . . . 6  |-  ( y  e.  ( ( Base `  W )  ^m  ( Base `  W ) )  <-> 
( x  e.  _V  /\  y  e.  ( (
Base `  W )  ^m  ( Base `  W
) ) ) )
2120anbi2i 698 . . . . 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 274 . . . 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 4424 . . 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 4420 . . 3  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
25 inopab 4920 . . 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 2454 . 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 2454 1  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3016    i^i cin 3371   {copab 4417    |-> cmpt 4418    X. cxp 4787   dom cdm 4789   -->wf 5533   ` cfv 5537  (class class class)co 6242    ^m cmap 7420   Basecbs 15057   proj1cpj1 17223   LSubSpclss 18091   ocvcocv 19158   projcpj 19198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-rab 2717  df-v 3018  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-fv 5545  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-map 7422  df-pj 19201
This theorem is referenced by:  pjval  19208  pjff  19210
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