Theorem pclvalN 36011

Description: Value of 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
pclvalN	|- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. S | X C_ y } )

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

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

Proof of Theorem pclvalN

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pclfval.a	. . . 4 |- A = ( Atoms ` K )
2		fvex 5858	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2538	. . 3 |- A e. _V
4	3	elpw2 4601	. 2 |- ( X e. ~P A <-> X C_ A )
5		pclfval.s	. . . . . 6 |- S = ( PSubSp ` K )
6		pclfval.c	. . . . . 6 |- U = ( PCl ` K )
7	1, 5, 6	pclfvalN 36010	. . . . 5 |- ( K e. V -> U = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )
8	7	fveq1d 5850	. . . 4 |- ( K e. V -> ( U ` X ) = ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) )
9	8	adantr 463	. . 3 |- ( ( K e. V /\ X e. ~P A ) -> ( U ` X ) = ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) )
10		simpr 459	. . . 4 |- ( ( K e. V /\ X e. ~P A ) -> X e. ~P A )
11		elpwi 4008	. . . . . . . 8 |- ( X e. ~P A -> X C_ A )
12	11	adantl 464	. . . . . . 7 |- ( ( K e. V /\ X e. ~P A ) -> X C_ A )
13	1, 5	atpsubN 35874	. . . . . . . . 9 |- ( K e. V -> A e. S )
14	13	adantr 463	. . . . . . . 8 |- ( ( K e. V /\ X e. ~P A ) -> A e. S )
15		sseq2 3511	. . . . . . . . 9 |- ( y = A -> ( X C_ y <-> X C_ A ) )
16	15	elrab3 3255	. . . . . . . 8 |- ( A e. S -> ( A e. { y e. S | X C_ y } <-> X C_ A ) )
17	14, 16	syl 16	. . . . . . 7 |- ( ( K e. V /\ X e. ~P A ) -> ( A e. { y e. S | X C_ y } <-> X C_ A ) )
18	12, 17	mpbird 232	. . . . . 6 |- ( ( K e. V /\ X e. ~P A ) -> A e. { y e. S | X C_ y } )
19		ne0i 3789	. . . . . 6 |- ( A e. { y e. S | X C_ y } -> { y e. S | X C_ y } =/= (/) )
20	18, 19	syl 16	. . . . 5 |- ( ( K e. V /\ X e. ~P A ) -> { y e. S | X C_ y } =/= (/) )
21		intex 4593	. . . . 5 |- ( { y e. S | X C_ y } =/= (/) <-> |^| { y e. S | X C_ y } e. _V )
22	20, 21	sylib 196	. . . 4 |- ( ( K e. V /\ X e. ~P A ) -> |^| { y e. S | X C_ y } e. _V )
23		sseq1 3510	. . . . . . 7 |- ( x = X -> ( x C_ y <-> X C_ y ) )
24	23	rabbidv 3098	. . . . . 6 |- ( x = X -> { y e. S | x C_ y } = { y e. S | X C_ y } )
25	24	inteqd 4276	. . . . 5 |- ( x = X -> |^| { y e. S | x C_ y } = |^| { y e. S | X C_ y } )
26		eqid 2454	. . . . 5 |- ( x e. ~P A |-> |^| { y e. S | x C_ y } ) = ( x e. ~P A |-> |^| { y e. S | x C_ y } )
27	25, 26	fvmptg 5929	. . . 4 |- ( ( X e. ~P A /\ |^| { y e. S | X C_ y } e. _V ) -> ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) = |^| { y e. S | X C_ y } )
28	10, 22, 27	syl2anc 659	. . 3 |- ( ( K e. V /\ X e. ~P A ) -> ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) = |^| { y e. S | X C_ y } )
29	9, 28	eqtrd 2495	. 2 |- ( ( K e. V /\ X e. ~P A ) -> ( U ` X ) = |^| { y e. S | X C_ y } )
30	4, 29	sylan2br 474	1 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. S | X C_ y } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   =/= wne 2649   {crab 2808   _Vcvv 3106   C_ wss 3461   (/)c0 3783   ~Pcpw 3999   |^|cint 4271   |-> cmpt 4497   ` cfv 5570   Atomscatm 35385   PSubSpcpsubsp 35617   PClcpclN 36008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-psubsp 35624  df-pclN 36009

This theorem is referenced by:  pclclN  36012  elpclN  36013  elpcliN  36014  pclssN  36015  pclssidN  36016  pclidN  36017