Theorem dicelval3 36469

Description: Member of the partial isomorphism C. (Contributed by NM, 26-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 )
dicval2.g	|- G = ( iota_ g e. T ( g ` P ) = Q )

Assertion

Ref	Expression
dicelval3	|- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> E. s e. E Y = <. ( s ` G ) , s >. ) )

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

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

Proof of Theorem dicelval3

Dummy variable f is 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		dicval2.g	. . . 4 |- G = ( iota_ g e. T ( g ` P ) = Q )
9	1, 2, 3, 4, 5, 6, 7, 8	dicval2 36468	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( I ` Q ) = { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } )
10	9	eleq2d 2687	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } ) )
11		excom 2042	. . . 4 |- ( E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) )
12		an12 838	. . . . . . 7 |- ( ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) )
13	12	exbii 1774	. . . . . 6 |- ( E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. f ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) )
14		fvex 6201	. . . . . . 7 |- ( s ` G ) e. _V
15		opeq1 4402	. . . . . . . . 9 |- ( f = ( s ` G ) -> <. f , s >. = <. ( s ` G ) , s >. )
16	15	eqeq2d 2632	. . . . . . . 8 |- ( f = ( s ` G ) -> ( Y = <. f , s >. <-> Y = <. ( s ` G ) , s >. ) )
17	16	anbi1d 741	. . . . . . 7 |- ( f = ( s ` G ) -> ( ( Y = <. f , s >. /\ s e. E ) <-> ( Y = <. ( s ` G ) , s >. /\ s e. E ) ) )
18	14, 17	ceqsexv 3242	. . . . . 6 |- ( E. f ( f = ( s ` G ) /\ ( Y = <. f , s >. /\ s e. E ) ) <-> ( Y = <. ( s ` G ) , s >. /\ s e. E ) )
19		ancom 466	. . . . . 6 |- ( ( Y = <. ( s ` G ) , s >. /\ s e. E ) <-> ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
20	13, 18, 19	3bitri 286	. . . . 5 |- ( E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
21	20	exbii 1774	. . . 4 |- ( E. s E. f ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
22	11, 21	bitri 264	. . 3 |- ( E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
23		elopab 4983	. . 3 |- ( Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } <-> E. f E. s ( Y = <. f , s >. /\ ( f = ( s ` G ) /\ s e. E ) ) )
24		df-rex 2918	. . 3 |- ( E. s e. E Y = <. ( s ` G ) , s >. <-> E. s ( s e. E /\ Y = <. ( s ` G ) , s >. ) )
25	22, 23, 24	3bitr4i 292	. 2 |- ( Y e. { <. f , s >. | ( f = ( s ` G ) /\ s e. E ) } <-> E. s e. E Y = <. ( s ` G ) , s >. )
26	10, 25	syl6bb 276	1 |- ( ( ( K e. V /\ W e. H ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( Y e. ( I ` Q ) <-> E. s e. E Y = <. ( s ` G ) , s >. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   <.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:  cdlemn11pre  36499  dihord2pre  36514