Theorem dicvalrelN 31668

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 4960	. . . 4 |- Rel { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) }
2		eqid 2404	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
3		eqid 2404	. . . . . . . . . 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 31667	. . . . . . . . 9 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. ( Atoms ` K ) | -. p ( le ` K ) W } )
7	6	eleq2d 2471	. . . . . . . 8 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } ) )
8		breq1 4175	. . . . . . . . . 10 |- ( p = X -> ( p ( le ` K ) W <-> X ( le ` K ) W ) )
9	8	notbid 286	. . . . . . . . 9 |- ( p = X -> ( -. p ( le ` K ) W <-> -. X ( le ` K ) W ) )
10	9	elrab 3052	. . . . . . . 8 |- ( X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
11	7, 10	syl6bb 253	. . . . . . 7 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) ) )
12	11	biimpa 471	. . . . . 6 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
13		eqid 2404	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
14		eqid 2404	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
15		eqid 2404	. . . . . . 7 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16	2, 3, 4, 13, 14, 15, 5	dicval 31659	. . . . . 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 457	. . . . 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 4920	. . . 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 225	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
20	19	ex 424	. 2 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I -> Rel ( I ` X ) ) )
21		rel0 4958	. . 3 |- Rel (/)
22		ndmfv 5714	. . . 4 |- ( -. X e. dom I -> ( I ` X ) = (/) )
23	22	releqd 4920	. . 3 |- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
24	21, 23	mpbiri 225	. 2 |- ( -. X e. dom I -> Rel ( I ` X ) )
25	20, 24	pm2.61d1 153	1 |- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   {crab 2670   (/)c0 3588   class class class wbr 4172   {copab 4225   dom cdm 4837   Rel wrel 4842   ` cfv 5413   iota_crio 6501   lecple 13491   occoc 13492   Atomscatm 29746   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234   DIsoCcdic 31655

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6508  df-dic 31656