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Theorem pjfval2 18132
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 4350 . . 3  |-  ( x  e.  ( LSubSp `  W
)  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  ( LSubSp `  W )  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
2 df-xp 4844 . . 3  |-  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) )  =  { <. x ,  y >.  |  ( x  e.  _V  /\  y  e.  ( ( Base `  W )  ^m  ( Base `  W )
) ) }
31, 2ineq12i 3548 . 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 2441 . . 3  |-  ( Base `  W )  =  (
Base `  W )
5 eqid 2441 . . 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 18129 . 2  |-  K  =  ( ( x  e.  ( LSubSp `  W )  |->  ( x P ( 
._|_  `  x ) ) )  i^i  ( _V 
X.  ( ( Base `  W )  ^m  ( Base `  W ) ) ) )
104, 5, 6, 7, 8pjdm 18130 . . . . . . 7  |-  ( x  e.  dom  K  <->  ( x  e.  ( LSubSp `  W )  /\  ( x P ( 
._|_  `  x ) ) : ( Base `  W
) --> ( Base `  W
) ) )
11 eleq1 2501 . . . . . . . . 9  |-  ( y  =  ( x P (  ._|_  `  x ) )  ->  ( y  e.  ( ( Base `  W
)  ^m  ( Base `  W ) )  <->  ( x P (  ._|_  `  x
) )  e.  ( ( Base `  W
)  ^m  ( Base `  W ) ) ) )
12 fvex 5699 . . . . . . . . . 10  |-  ( Base `  W )  e.  _V
1312, 12elmap 7239 . . . . . . . . 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 796 . . . . 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 2973 . . . . . . 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 4354 . . 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 4350 . . 3  |-  ( x  e.  dom  K  |->  ( x P (  ._|_  `  x ) ) )  =  { <. x ,  y >.  |  ( x  e.  dom  K  /\  y  =  (
x P (  ._|_  `  x ) ) ) }
25 inopab 4968 . . 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 2471 . 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 2471 1  |-  K  =  ( x  e.  dom  K 
|->  ( x P ( 
._|_  `  x ) ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2970    i^i cin 3325   {copab 4347    e. cmpt 4348    X. cxp 4836   dom cdm 4838   -->wf 5412   ` cfv 5416  (class class class)co 6089    ^m cmap 7212   Basecbs 14172   proj1cpj1 16132   LSubSpclss 17011   ocvcocv 18083   projcpj 18123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-pj 18126
This theorem is referenced by:  pjval  18133  pjff  18135
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