Theorem diaelval 36322

Description: Member of the partial isomorphism A for a lattice K. (Contributed by NM, 3-Dec-2013.)

Hypotheses

Ref	Expression
diaval.b	|- B = ( Base ` K )
diaval.l	|- .<_ = ( le ` K )
diaval.h	|- H = ( LHyp ` K )
diaval.t	|- T = ( ( LTrn ` K ) ` W )
diaval.r	|- R = ( ( trL ` K ) ` W )
diaval.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
diaelval	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )

Proof of Theorem diaelval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		diaval.b	. . . 4 |- B = ( Base ` K )
2		diaval.l	. . . 4 |- .<_ = ( le ` K )
3		diaval.h	. . . 4 |- H = ( LHyp ` K )
4		diaval.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
5		diaval.r	. . . 4 |- R = ( ( trL ` K ) ` W )
6		diaval.i	. . . 4 |- I = ( ( DIsoA ` K ) ` W )
7	1, 2, 3, 4, 5, 6	diaval 36321	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. T | ( R ` f ) .<_ X } )
8	7	eleq2d 2687	. 2 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> F e. { f e. T | ( R ` f ) .<_ X } ) )
9		fveq2 6191	. . . 4 |- ( f = F -> ( R ` f ) = ( R ` F ) )
10	9	breq1d 4663	. . 3 |- ( f = F -> ( ( R ` f ) .<_ X <-> ( R ` F ) .<_ X ) )
11	10	elrab 3363	. 2 |- ( F e. { f e. T | ( R ` f ) .<_ X } <-> ( F e. T /\ ( R ` F ) .<_ X ) )
12	8, 11	syl6bb 276	1 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoAcdia 36317

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-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-pr 4906

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-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-disoa 36318

This theorem is referenced by:  dian0  36328  diatrl  36333  dialss  36335  diaglbN  36344  dibelval3  36436  dibopelval3  36437  diblss  36459