Theorem dibelval2nd 36441

Description: Membership in value of the partial isomorphism B for a lattice K. (Contributed by NM, 13-Feb-2014.)

Hypotheses

Ref	Expression
dibelval2nd.b	|- B = ( Base ` K )
dibelval2nd.l	|- .<_ = ( le ` K )
dibelval2nd.h	|- H = ( LHyp ` K )
dibelval2nd.t	|- T = ( ( LTrn ` K ) ` W )
dibelval2nd.o	|- .0. = ( f e. T |-> ( _I |` B ) )
dibelval2nd.i	|- I = ( ( DIsoB ` K ) ` W )

Assertion

Ref	Expression
dibelval2nd	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )

Distinct variable groups:   f, K   f, W

Allowed substitution hints:   B( f)   T( f)   H( f)   I( f)   .<_ ( f)   V( f)   X( f)   Y( f)   .0. ( f)

Proof of Theorem dibelval2nd

Step	Hyp	Ref	Expression
1		dibelval2nd.b	. . . . 5 |- B = ( Base ` K )
2		dibelval2nd.l	. . . . 5 |- .<_ = ( le ` K )
3		dibelval2nd.h	. . . . 5 |- H = ( LHyp ` K )
4		dibelval2nd.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
5		dibelval2nd.o	. . . . 5 |- .0. = ( f e. T |-> ( _I |` B ) )
6		eqid 2622	. . . . 5 |- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
7		dibelval2nd.i	. . . . 5 |- I = ( ( DIsoB ` K ) ` W )
8	1, 2, 3, 4, 5, 6, 7	dibval2 36433	. . . 4 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
9	8	eleq2d 2687	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( Y e. ( I ` X ) <-> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) ) )
10	9	biimp3a 1432	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
11		xp2nd 7199	. 2 |- ( Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) -> ( 2nd ` Y ) e. { .0. } )
12		elsni 4194	. 2 |- ( ( 2nd ` Y ) e. { .0. } -> ( 2nd ` Y ) = .0. )
13	10, 11, 12	3syl 18	1 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   |` cres 5116   ` cfv 5888   2ndc2nd 7167   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   DIsoAcdia 36317   DIsoBcdib 36427

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-2nd 7169  df-disoa 36318  df-dib 36428

This theorem is referenced by:  diblss  36459