Theorem dicvalrelN 34491

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 2420	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
3		eqid 2420	. . . . . . . . . 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 34490	. . . . . . . . 9 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. ( Atoms ` K ) | -. p ( le ` K ) W } )
7	6	eleq2d 2490	. . . . . . . 8 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } ) )
8		breq1 4420	. . . . . . . . . 10 |- ( p = X -> ( p ( le ` K ) W <-> X ( le ` K ) W ) )
9	8	notbid 295	. . . . . . . . 9 |- ( p = X -> ( -. p ( le ` K ) W <-> -. X ( le ` K ) W ) )
10	9	elrab 3226	. . . . . . . 8 |- ( X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
11	7, 10	syl6bb 264	. . . . . . 7 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) ) )
12	11	biimpa 486	. . . . . 6 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
13		eqid 2420	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
14		eqid 2420	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
15		eqid 2420	. . . . . . 7 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16	2, 3, 4, 13, 14, 15, 5	dicval 34482	. . . . . 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 472	. . . . 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 4930	. . . 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 236	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
20	19	ex 435	. 2 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I -> Rel ( I ` X ) ) )
21		rel0 4969	. . 3 |- Rel (/)
22		ndmfv 5896	. . . 4 |- ( -. X e. dom I -> ( I ` X ) = (/) )
23	22	releqd 4930	. . 3 |- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
24	21, 23	mpbiri 236	. 2 |- ( -. X e. dom I -> Rel ( I ` X ) )
25	20, 24	pm2.61d1 162	1 |- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1867   {crab 2777   (/)c0 3758   class class class wbr 4417   {copab 4474   dom cdm 4845   Rel wrel 4850   ` cfv 5592   iota_crio 6257   lecple 15149   occoc 15150   Atomscatm 32567   LHypclh 33287   LTrncltrn 33404   TEndoctendo 34057   DIsoCcdic 34478

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  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 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-dic 34479

This theorem is referenced by: (None)