Theorem dicelval1sta 36476

Description: Membership in value of the partial isomorphism C for a lattice K. (Contributed by NM, 16-Feb-2014.)

Hypotheses

Ref	Expression
dicelval1sta.l	|- .<_ = ( le ` K )
dicelval1sta.a	|- A = ( Atoms ` K )
dicelval1sta.h	|- H = ( LHyp ` K )
dicelval1sta.p	|- P = ( ( oc ` K ) ` W )
dicelval1sta.t	|- T = ( ( LTrn ` K ) ` W )
dicelval1sta.i	|- I = ( ( DIsoC ` K ) ` W )

Assertion

Ref	Expression
dicelval1sta	|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) )

Distinct variable groups:   g, K   Q, g   T, g   g, W

Allowed substitution hints:   A( g)   P( g)   H( g)   I( g)   .<_ ( g)   V( g)   Y( g)

Proof of Theorem dicelval1sta

Dummy variables f s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicelval1sta.l	. . . . . 6 |- .<_ = ( le ` K )
2		dicelval1sta.a	. . . . . 6 |- A = ( Atoms ` K )
3		dicelval1sta.h	. . . . . 6 |- H = ( LHyp ` K )
4		dicelval1sta.p	. . . . . 6 |- P = ( ( oc ` K ) ` W )
5		dicelval1sta.t	. . . . . 6 |- T = ( ( LTrn ` K ) ` W )
6		eqid 2622	. . . . . 6 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
7		dicelval1sta.i	. . . . . 6 |- I = ( ( DIsoC ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	dicval 36465	. . . . 5 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } )
9	8	eleq2d 2687	. . . 4 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> Y e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) )
10	9	biimp3a 1432	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> Y e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } )
11		eqeq1 2626	. . . . 5 |- ( f = ( 1st ` Y ) -> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> ( 1st ` Y ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
12	11	anbi1d 741	. . . 4 |- ( f = ( 1st ` Y ) -> ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) <-> ( ( 1st ` Y ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) ) )
13		fveq1 6190	. . . . . 6 |- ( s = ( 2nd ` Y ) -> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) )
14	13	eqeq2d 2632	. . . . 5 |- ( s = ( 2nd ` Y ) -> ( ( 1st ` Y ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
15		eleq1 2689	. . . . 5 |- ( s = ( 2nd ` Y ) -> ( s e. ( ( TEndo ` K ) ` W ) <-> ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) )
16	14, 15	anbi12d 747	. . . 4 |- ( s = ( 2nd ` Y ) -> ( ( ( 1st ` Y ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) <-> ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) ) )
17	12, 16	elopabi 7231	. . 3 |- ( Y e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } -> ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) )
18	10, 17	syl 17	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ ( 2nd ` Y ) e. ( ( TEndo ` K ) ` W ) ) )
19	18	simpld 475	1 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) /\ Y e. ( I ` Q ) ) -> ( 1st ` Y ) = ( ( 2nd ` Y ) ` ( iota_ g e. T ( g ` P ) = Q ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712   ` cfv 5888   iota_crio 6610   1stc1st 7166   2ndc2nd 7167   lecple 15948   occoc 15949   Atomscatm 34550   LHypclh 35270   LTrncltrn 35387   TEndoctendo 36040   DIsoCcdic 36461

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  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-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-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-riota 6611  df-1st 7168  df-2nd 7169  df-dic 36462

This theorem is referenced by:  dicvaddcl  36479  dicvscacl  36480