Theorem pjfval2 20053

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.

Step	Hyp	Ref	Expression
1		df-mpt 4730	. . 3 |- ( x e. ( LSubSp ` W ) |-> ( x P ( ._|_ ` x ) ) ) = { <. x , y >. | ( x e. ( LSubSp ` W ) /\ y = ( x P ( ._|_ ` x ) ) ) }
2		df-xp 5120	. . 3 |- ( _V X. ( ( Base ` W ) ^m ( Base ` W ) ) ) = { <. x , y >. | ( x e. _V /\ y e. ( ( Base ` W ) ^m ( Base ` W ) ) ) }
3	1, 2	ineq12i 3812	. 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 2622	. . 3 |- ( Base ` W ) = ( Base ` W )
5		eqid 2622	. . 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 )
9	4, 5, 6, 7, 8	pjfval 20050	. 2 |- K = ( ( x e. ( LSubSp ` W ) |-> ( x P ( ._|_ ` x ) ) ) i^i ( _V X. ( ( Base ` W ) ^m ( Base ` W ) ) ) )
10	4, 5, 6, 7, 8	pjdm 20051	. . . . . . 7 |- ( x e. dom K <-> ( x e. ( LSubSp ` W ) /\ ( x P ( ._|_ ` x ) ) : ( Base ` W ) --> ( Base ` W ) ) )
11		eleq1 2689	. . . . . . . . 9 |- ( y = ( x P ( ._|_ ` x ) ) -> ( y e. ( ( Base ` W ) ^m ( Base ` W ) ) <-> ( x P ( ._|_ ` x ) ) e. ( ( Base ` W ) ^m ( Base ` W ) ) ) )
12		fvex 6201	. . . . . . . . . 10 |- ( Base ` W ) e. _V
13	12, 12	elmap 7886	. . . . . . . . 9 |- ( ( x P ( ._|_ ` x ) ) e. ( ( Base ` W ) ^m ( Base ` W ) ) <-> ( x P ( ._|_ ` x ) ) : ( Base ` W ) --> ( Base ` W ) )
14	11, 13	syl6rbb 277	. . . . . . . 8 |- ( y = ( x P ( ._|_ ` x ) ) -> ( ( x P ( ._|_ ` x ) ) : ( Base ` W ) --> ( Base ` W ) <-> y e. ( ( Base ` W ) ^m ( Base ` W ) ) ) )
15	14	anbi2d 740	. . . . . . 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 ) ) ) ) )
16	10, 15	syl5bb 272	. . . . . 6 |- ( y = ( x P ( ._|_ ` x ) ) -> ( x e. dom K <-> ( x e. ( LSubSp ` W ) /\ y e. ( ( Base ` W ) ^m ( Base ` W ) ) ) ) )
17	16	pm5.32ri 670	. . . . 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 839	. . . . 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 3203	. . . . . . 7 |- x e. _V
20	19	biantrur 527	. . . . . 6 |- ( y e. ( ( Base ` W ) ^m ( Base ` W ) ) <-> ( x e. _V /\ y e. ( ( Base ` W ) ^m ( Base ` W ) ) ) )
21	20	anbi2i 730	. . . . 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 ) ) ) ) )
22	17, 18, 21	3bitri 286	. . . 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 ) ) ) ) )
23	22	opabbii 4717	. . 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 4730	. . 3 |- ( x e. dom K |-> ( x P ( ._|_ ` x ) ) ) = { <. x , y >. | ( x e. dom K /\ y = ( x P ( ._|_ ` x ) ) ) }
25		inopab 5252	. . 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 ) ) ) ) }
26	23, 24, 25	3eqtr4i 2654	. 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 ) ) ) } )
27	3, 9, 26	3eqtr4i 2654	1 |- K = ( x e. dom K |-> ( x P ( ._|_ ` x ) ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   proj1cpj1 18050   LSubSpclss 18932   ocvcocv 20004   projcpj 20044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-pj 20047

This theorem is referenced by:  pjval  20054  pjff  20056