Theorem dicvalrelN 34835

Description: The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dicvalrel.h	|- H = ( LHyp ` K )
dicvalrel.i	|- I = ( ( DIsoC ` K ) ` W )

Assertion

Ref	Expression
dicvalrelN	|- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Proof of Theorem dicvalrelN

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

Step	Hyp	Ref	Expression
1		relopab 4971	. . . 4 |- Rel { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) }
2		eqid 2443	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
3		eqid 2443	. . . . . . . . . 10 |- ( Atoms ` K ) = ( Atoms ` K )
4		dicvalrel.h	. . . . . . . . . 10 |- H = ( LHyp ` K )
5		dicvalrel.i	. . . . . . . . . 10 |- I = ( ( DIsoC ` K ) ` W )
6	2, 3, 4, 5	dicdmN 34834	. . . . . . . . 9 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. ( Atoms ` K ) | -. p ( le ` K ) W } )
7	6	eleq2d 2510	. . . . . . . 8 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } ) )
8		breq1 4300	. . . . . . . . . 10 |- ( p = X -> ( p ( le ` K ) W <-> X ( le ` K ) W ) )
9	8	notbid 294	. . . . . . . . 9 |- ( p = X -> ( -. p ( le ` K ) W <-> -. X ( le ` K ) W ) )
10	9	elrab 3122	. . . . . . . 8 |- ( X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
11	7, 10	syl6bb 261	. . . . . . 7 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) ) )
12	11	biimpa 484	. . . . . 6 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
13		eqid 2443	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
14		eqid 2443	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
15		eqid 2443	. . . . . . 7 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16	2, 3, 4, 13, 14, 15, 5	dicval 34826	. . . . . 6 |- ( ( ( K e. V /\ W e. H ) /\ ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) ) -> ( I ` X ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } )
17	12, 16	syldan 470	. . . . 5 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) = { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } )
18	17	releqd 4929	. . . 4 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( Rel ( I ` X ) <-> Rel { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) )
19	1, 18	mpbiri 233	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
20	19	ex 434	. 2 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I -> Rel ( I ` X ) ) )
21		rel0 4969	. . 3 |- Rel (/)
22		ndmfv 5719	. . . 4 |- ( -. X e. dom I -> ( I ` X ) = (/) )
23	22	releqd 4929	. . 3 |- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
24	21, 23	mpbiri 233	. 2 |- ( -. X e. dom I -> Rel ( I ` X ) )
25	20, 24	pm2.61d1 159	1 |- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   {crab 2724   (/)c0 3642   class class class wbr 4297   {copab 4354   dom cdm 4845   Rel wrel 4850   ` cfv 5423   iota_crio 6056   lecple 14250   occoc 14251   Atomscatm 32913   LHypclh 33633   LTrncltrn 33750   TEndoctendo 34401   DIsoCcdic 34822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-dic 34823

This theorem is referenced by: (None)