Theorem dicvalrelN 36000

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 5129	. . . 4 |- Rel { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) }
2		eqid 2467	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
3		eqid 2467	. . . . . . . . . 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 35999	. . . . . . . . 9 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. ( Atoms ` K ) | -. p ( le ` K ) W } )
7	6	eleq2d 2537	. . . . . . . 8 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } ) )
8		breq1 4450	. . . . . . . . . 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 3261	. . . . . . . 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 2467	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
14		eqid 2467	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
15		eqid 2467	. . . . . . 7 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16	2, 3, 4, 13, 14, 15, 5	dicval 35991	. . . . . 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 5087	. . . 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 5127	. . 3 |- Rel (/)
22		ndmfv 5890	. . . 4 |- ( -. X e. dom I -> ( I ` X ) = (/) )
23	22	releqd 5087	. . 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 1379   e. wcel 1767   {crab 2818   (/)c0 3785   class class class wbr 4447   {copab 4504   dom cdm 4999   Rel wrel 5004   ` cfv 5588   iota_crio 6244   lecple 14562   occoc 14563   Atomscatm 34078   LHypclh 34798   LTrncltrn 34915   TEndoctendo 35566   DIsoCcdic 35987

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-dic 35988

This theorem is referenced by: (None)