Theorem pclvalN 33455

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 5875	. . . 4 |- ( Atoms ` K ) e. _V
3	1, 2	eqeltri 2525	. . 3 |- A e. _V
4	3	elpw2 4567	. 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 33454	. . . . 5 |- ( K e. V -> U = ( x e. ~P A |-> |^| { y e. S | x C_ y } ) )
8	7	fveq1d 5867	. . . 4 |- ( K e. V -> ( U ` X ) = ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) )
9	8	adantr 467	. . 3 |- ( ( K e. V /\ X e. ~P A ) -> ( U ` X ) = ( ( x e. ~P A |-> |^| { y e. S | x C_ y } ) ` X ) )
10		simpr 463	. . . 4 |- ( ( K e. V /\ X e. ~P A ) -> X e. ~P A )
11		elpwi 3960	. . . . . . . 8 |- ( X e. ~P A -> X C_ A )
12	11	adantl 468	. . . . . . 7 |- ( ( K e. V /\ X e. ~P A ) -> X C_ A )
13	1, 5	atpsubN 33318	. . . . . . . . 9 |- ( K e. V -> A e. S )
14	13	adantr 467	. . . . . . . 8 |- ( ( K e. V /\ X e. ~P A ) -> A e. S )
15		sseq2 3454	. . . . . . . . 9 |- ( y = A -> ( X C_ y <-> X C_ A ) )
16	15	elrab3 3197	. . . . . . . 8 |- ( A e. S -> ( A e. { y e. S | X C_ y } <-> X C_ A ) )
17	14, 16	syl 17	. . . . . . 7 |- ( ( K e. V /\ X e. ~P A ) -> ( A e. { y e. S | X C_ y } <-> X C_ A ) )
18	12, 17	mpbird 236	. . . . . 6 |- ( ( K e. V /\ X e. ~P A ) -> A e. { y e. S | X C_ y } )
19		ne0i 3737	. . . . . 6 |- ( A e. { y e. S | X C_ y } -> { y e. S | X C_ y } =/= (/) )
20	18, 19	syl 17	. . . . 5 |- ( ( K e. V /\ X e. ~P A ) -> { y e. S | X C_ y } =/= (/) )
21		intex 4559	. . . . 5 |- ( { y e. S | X C_ y } =/= (/) <-> |^| { y e. S | X C_ y } e. _V )
22	20, 21	sylib 200	. . . 4 |- ( ( K e. V /\ X e. ~P A ) -> |^| { y e. S | X C_ y } e. _V )
23		sseq1 3453	. . . . . . 7 |- ( x = X -> ( x C_ y <-> X C_ y ) )
24	23	rabbidv 3036	. . . . . 6 |- ( x = X -> { y e. S | x C_ y } = { y e. S | X C_ y } )
25	24	inteqd 4239	. . . . 5 |- ( x = X -> |^| { y e. S | x C_ y } = |^| { y e. S | X C_ y } )
26		eqid 2451	. . . . 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 5946	. . . 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 667	. . 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 2485	. 2 |- ( ( K e. V /\ X e. ~P A ) -> ( U ` X ) = |^| { y e. S | X C_ y } )
30	4, 29	sylan2br 479	1 |- ( ( K e. V /\ X C_ A ) -> ( U ` X ) = |^| { y e. S | X C_ y } )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   {crab 2741   _Vcvv 3045   C_ wss 3404   (/)c0 3731   ~Pcpw 3951   |^|cint 4234   |-> cmpt 4461   ` cfv 5582   Atomscatm 32829   PSubSpcpsubsp 33061   PClcpclN 33452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-psubsp 33068  df-pclN 33453

This theorem is referenced by:  pclclN  33456  elpclN  33457  elpcliN  33458  pclssN  33459  pclssidN  33460  pclidN  33461