Theorem dicopelval 36466

Description: Membership in value of the partial isomorphism C for a lattice K. (Contributed by NM, 15-Feb-2014.)

Hypotheses

Ref	Expression
dicval.l	|- .<_ = ( le ` K )
dicval.a	|- A = ( Atoms ` K )
dicval.h	|- H = ( LHyp ` K )
dicval.p	|- P = ( ( oc ` K ) ` W )
dicval.t	|- T = ( ( LTrn ` K ) ` W )
dicval.e	|- E = ( ( TEndo ` K ) ` W )
dicval.i	|- I = ( ( DIsoC ` K ) ` W )
dicelval.f	|- F e. _V
dicelval.s	|- S e. _V

Assertion

Ref	Expression
dicopelval	|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )

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

Allowed substitution hints:   A( g)   P( g)   S( g)   E( g)   F( g)   H( g)   I( g)   .<_ ( g)   V( g)

Proof of Theorem dicopelval

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

Step	Hyp	Ref	Expression
1		dicval.l	. . . 4 |- .<_ = ( le ` K )
2		dicval.a	. . . 4 |- A = ( Atoms ` K )
3		dicval.h	. . . 4 |- H = ( LHyp ` K )
4		dicval.p	. . . 4 |- P = ( ( oc ` K ) ` W )
5		dicval.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
6		dicval.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
7		dicval.i	. . . 4 |- I = ( ( DIsoC ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	dicval 36465	. . 3 |- ( ( ( 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. E ) } )
9	8	eleq2d 2687	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> <. F , S >. e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } ) )
10		dicelval.f	. . 3 |- F e. _V
11		dicelval.s	. . 3 |- S e. _V
12		eqeq1 2626	. . . 4 |- ( f = F -> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
13	12	anbi1d 741	. . 3 |- ( f = F -> ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) <-> ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) ) )
14		fveq1 6190	. . . . 5 |- ( s = S -> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) )
15	14	eqeq2d 2632	. . . 4 |- ( s = S -> ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) <-> F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
16		eleq1 2689	. . . 4 |- ( s = S -> ( s e. E <-> S e. E ) )
17	15, 16	anbi12d 747	. . 3 |- ( s = S -> ( ( F = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )
18	10, 11, 13, 17	opelopab 4997	. 2 |- ( <. F , S >. e. { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) )
19	9, 18	syl6bb 276	1 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( <. F , S >. e. ( I ` Q ) <-> ( F = ( S ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ S e. E ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {copab 4712   ` cfv 5888   iota_crio 6610   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-dic 36462

This theorem is referenced by:  dicopelval2  36470  dicvaddcl  36479  dicvscacl  36480  dicn0  36481