Theorem dicval 36465

Description: The partial isomorphism C for a lattice K. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)

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 )

Assertion

Ref	Expression
dicval	|- ( ( ( 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 ) } )

Distinct variable groups:   f, g, s, K   T, g   f, W, g, s   f, E, s   P, f   Q, f, g, s   T, f

Allowed substitution hints:   A( f, g, s)   P( g, s)   T( s)   E( g)   H( f, g, s)   I( f, g, s)   .<_ ( f, g, s)   V( f, g, s)

Proof of Theorem dicval

Dummy variables r q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicval.l	. . . . 5 |- .<_ = ( le ` K )
2		dicval.a	. . . . 5 |- A = ( Atoms ` K )
3		dicval.h	. . . . 5 |- H = ( LHyp ` K )
4		dicval.p	. . . . 5 |- P = ( ( oc ` K ) ` W )
5		dicval.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
6		dicval.e	. . . . 5 |- E = ( ( TEndo ` K ) ` W )
7		dicval.i	. . . . 5 |- I = ( ( DIsoC ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	dicfval 36464	. . . 4 |- ( ( K e. V /\ W e. H ) -> I = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
9	8	adantr 481	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> I = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) )
10	9	fveq1d 6193	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = ( ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) ` Q ) )
11		simpr 477	. . . 4 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Q e. A /\ -. Q .<_ W ) )
12		breq1 4656	. . . . . 6 |- ( r = Q -> ( r .<_ W <-> Q .<_ W ) )
13	12	notbid 308	. . . . 5 |- ( r = Q -> ( -. r .<_ W <-> -. Q .<_ W ) )
14	13	elrab 3363	. . . 4 |- ( Q e. { r e. A | -. r .<_ W } <-> ( Q e. A /\ -. Q .<_ W ) )
15	11, 14	sylibr 224	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> Q e. { r e. A | -. r .<_ W } )
16		eqeq2 2633	. . . . . . . . 9 |- ( q = Q -> ( ( g ` P ) = q <-> ( g ` P ) = Q ) )
17	16	riotabidv 6613	. . . . . . . 8 |- ( q = Q -> ( iota_ g e. T ( g ` P ) = q ) = ( iota_ g e. T ( g ` P ) = Q ) )
18	17	fveq2d 6195	. . . . . . 7 |- ( q = Q -> ( s ` ( iota_ g e. T ( g ` P ) = q ) ) = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) )
19	18	eqeq2d 2632	. . . . . 6 |- ( q = Q -> ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) <-> f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) ) )
20	19	anbi1d 741	. . . . 5 |- ( q = Q -> ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) <-> ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) ) )
21	20	opabbidv 4716	. . . 4 |- ( q = Q -> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } )
22		eqid 2622	. . . 4 |- ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) = ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } )
23		fvex 6201	. . . . . . . . . . 11 |- ( ( TEndo ` K ) ` W ) e. _V
24	6, 23	eqeltri 2697	. . . . . . . . . 10 |- E e. _V
25	24	uniex 6953	. . . . . . . . 9 |- U. E e. _V
26	25	rnex 7100	. . . . . . . 8 |- ran U. E e. _V
27	26	uniex 6953	. . . . . . 7 |- U. ran U. E e. _V
28	27	pwex 4848	. . . . . 6 |- ~P U. ran U. E e. _V
29	28, 24	xpex 6962	. . . . 5 |- ( ~P U. ran U. E X. E ) e. _V
30		simpl 473	. . . . . . . . 9 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) )
31		fvssunirn 6217	. . . . . . . . . . 11 |- ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) C_ U. ran s
32		elssuni 4467	. . . . . . . . . . . . 13 |- ( s e. E -> s C_ U. E )
33	32	adantl 482	. . . . . . . . . . . 12 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> s C_ U. E )
34		rnss 5354	. . . . . . . . . . . 12 |- ( s C_ U. E -> ran s C_ ran U. E )
35		uniss 4458	. . . . . . . . . . . 12 |- ( ran s C_ ran U. E -> U. ran s C_ U. ran U. E )
36	33, 34, 35	3syl 18	. . . . . . . . . . 11 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> U. ran s C_ U. ran U. E )
37	31, 36	syl5ss 3614	. . . . . . . . . 10 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) C_ U. ran U. E )
38	27	elpw2 4828	. . . . . . . . . 10 |- ( ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) e. ~P U. ran U. E <-> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) C_ U. ran U. E )
39	37, 38	sylibr 224	. . . . . . . . 9 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) e. ~P U. ran U. E )
40	30, 39	eqeltrd 2701	. . . . . . . 8 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> f e. ~P U. ran U. E )
41		simpr 477	. . . . . . . 8 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> s e. E )
42	40, 41	jca 554	. . . . . . 7 |- ( ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) -> ( f e. ~P U. ran U. E /\ s e. E ) )
43	42	ssopab2i 5003	. . . . . 6 |- { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } C_ { <. f , s >. | ( f e. ~P U. ran U. E /\ s e. E ) }
44		df-xp 5120	. . . . . 6 |- ( ~P U. ran U. E X. E ) = { <. f , s >. | ( f e. ~P U. ran U. E /\ s e. E ) }
45	43, 44	sseqtr4i 3638	. . . . 5 |- { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } C_ ( ~P U. ran U. E X. E )
46	29, 45	ssexi 4803	. . . 4 |- { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } e. _V
47	21, 22, 46	fvmpt 6282	. . 3 |- ( Q e. { r e. A | -. r .<_ W } -> ( ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) ` Q ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } )
48	15, 47	syl 17	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( q e. { r e. A | -. r .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = q ) ) /\ s e. E ) } ) ` Q ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. T ( g ` P ) = Q ) ) /\ s e. E ) } )
49	10, 48	eqtrd 2656	1 |- ( ( ( 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 ) } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   ran crn 5115   ` 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:  dicopelval  36466  dicelvalN  36467  dicval2  36468  dicfnN  36472  dicvalrelN  36474  dicssdvh  36475  dicelval1sta  36476  dihpN  36625