Theorem elpcliN 35179

Description: Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
elpcli.s	|- S = ( PSubSp ` K )
elpcli.c	|- U = ( PCl ` K )

Assertion

Ref	Expression
elpcliN	|- ( ( ( K e. V /\ X C_ Y /\ Y e. S ) /\ Q e. ( U ` X ) ) -> Q e. Y )

Proof of Theorem elpcliN

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> K e. V )
2		simp2 1062	. . . . . . 7 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> X C_ Y )
3		eqid 2622	. . . . . . . . 9 |- ( Atoms ` K ) = ( Atoms ` K )
4		elpcli.s	. . . . . . . . 9 |- S = ( PSubSp ` K )
5	3, 4	psubssat 35040	. . . . . . . 8 |- ( ( K e. V /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
6	5	3adant2 1080	. . . . . . 7 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
7	2, 6	sstrd 3613	. . . . . 6 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> X C_ ( Atoms ` K ) )
8		elpcli.c	. . . . . . 7 |- U = ( PCl ` K )
9	3, 4, 8	pclvalN 35176	. . . . . 6 |- ( ( K e. V /\ X C_ ( Atoms ` K ) ) -> ( U ` X ) = |^| { z e. S | X C_ z } )
10	1, 7, 9	syl2anc 693	. . . . 5 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> ( U ` X ) = |^| { z e. S | X C_ z } )
11	10	eleq2d 2687	. . . 4 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> ( Q e. ( U ` X ) <-> Q e. |^| { z e. S | X C_ z } ) )
12		elintrabg 4489	. . . . 5 |- ( Q e. |^| { z e. S | X C_ z } -> ( Q e. |^| { z e. S | X C_ z } <-> A. z e. S ( X C_ z -> Q e. z ) ) )
13	12	ibi 256	. . . 4 |- ( Q e. |^| { z e. S | X C_ z } -> A. z e. S ( X C_ z -> Q e. z ) )
14	11, 13	syl6bi 243	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> ( Q e. ( U ` X ) -> A. z e. S ( X C_ z -> Q e. z ) ) )
15		sseq2 3627	. . . . . . . 8 |- ( z = Y -> ( X C_ z <-> X C_ Y ) )
16		eleq2 2690	. . . . . . . 8 |- ( z = Y -> ( Q e. z <-> Q e. Y ) )
17	15, 16	imbi12d 334	. . . . . . 7 |- ( z = Y -> ( ( X C_ z -> Q e. z ) <-> ( X C_ Y -> Q e. Y ) ) )
18	17	rspccv 3306	. . . . . 6 |- ( A. z e. S ( X C_ z -> Q e. z ) -> ( Y e. S -> ( X C_ Y -> Q e. Y ) ) )
19	18	com13 88	. . . . 5 |- ( X C_ Y -> ( Y e. S -> ( A. z e. S ( X C_ z -> Q e. z ) -> Q e. Y ) ) )
20	19	imp 445	. . . 4 |- ( ( X C_ Y /\ Y e. S ) -> ( A. z e. S ( X C_ z -> Q e. z ) -> Q e. Y ) )
21	20	3adant1 1079	. . 3 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> ( A. z e. S ( X C_ z -> Q e. z ) -> Q e. Y ) )
22	14, 21	syld 47	. 2 |- ( ( K e. V /\ X C_ Y /\ Y e. S ) -> ( Q e. ( U ` X ) -> Q e. Y ) )
23	22	imp 445	1 |- ( ( ( K e. V /\ X C_ Y /\ Y e. S ) /\ Q e. ( U ` X ) ) -> Q e. Y )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   |^|cint 4475   ` 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-8 1992  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-ov 6653  df-psubsp 34789  df-pclN 35174

This theorem is referenced by:  pclfinclN  35236