Theorem pclfvalN 35175

Description: The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pclfval.a	|- A = ( Atoms ` K )
pclfval.s	|- S = ( PSubSp ` K )
pclfval.c	|- U = ( PCl ` K )

Assertion

Ref	Expression
pclfvalN	|- ( K e. V -> U = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )

Distinct variable groups:   x, y, A   x, K, y   x, S, y

Allowed substitution hints:   U( x, y)   V( x, y)

Proof of Theorem pclfvalN

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2 |- ( K e. V -> K e. _V )
2		pclfval.c	. . 3 |- U = ( PCl ` K )
3		fveq2 6191	. . . . . . 7 |- ( k = K -> ( Atoms ` k ) = ( Atoms ` K ) )
4		pclfval.a	. . . . . . 7 |- A = ( Atoms ` K )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( k = K -> ( Atoms ` k ) = A )
6	5	pweqd 4163	. . . . 5 |- ( k = K -> ~P ( Atoms ` k ) = ~P A )
7		fveq2 6191	. . . . . . . 8 |- ( k = K -> ( PSubSp ` k ) = ( PSubSp ` K ) )
8		pclfval.s	. . . . . . . 8 |- S = ( PSubSp ` K )
9	7, 8	syl6eqr 2674	. . . . . . 7 |- ( k = K -> ( PSubSp ` k ) = S )
10	9	rabeqdv 3194	. . . . . 6 |- ( k = K -> { y e. ( PSubSp ` k ) | x C_ y } = { y e. S | x C_ y } )
11	10	inteqd 4480	. . . . 5 |- ( k = K -> |^| { y e. ( PSubSp ` k ) | x C_ y } = |^| { y e. S | x C_ y } )
12	6, 11	mpteq12dv 4733	. . . 4 |- ( k = K -> ( x e. ~P ( Atoms ` k ) |-> |^| { y e. ( PSubSp ` k ) | x C_ y } ) = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )
13		df-pclN 35174	. . . 4 |- PCl = ( k e. _V |-> ( x e. ~P ( Atoms ` k ) |-> |^| { y e. ( PSubSp ` k ) | x C_ y } ) )
14		fvex 6201	. . . . . . 7 |- ( Atoms ` K ) e. _V
15	4, 14	eqeltri 2697	. . . . . 6 |- A e. _V
16	15	pwex 4848	. . . . 5 |- ~P A e. _V
17	16	mptex 6486	. . . 4 |- ( x e. ~P A |-> |^| { y e. S | x C_ y } ) e. _V
18	12, 13, 17	fvmpt 6282	. . 3 |- ( K e. _V -> ( PCl ` K ) = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )
19	2, 18	syl5eq 2668	. 2 |- ( K e. _V -> U = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )
20	1, 19	syl 17	1 |- ( K e. V -> U = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |^|cint 4475   |-> cmpt 4729   ` cfv 5888   Atomscatm 34550   PSubSpcpsubsp 34782   PClcpclN 35173

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-int 4476  df-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-pclN 35174

This theorem is referenced by:  pclvalN  35176