Theorem dicvalrelN 34747

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 4959	. . . 4 |- Rel { <. f , s >. | ( f = ( s ` ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = X ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) }
2		eqid 2450	. . . . . . . . . 10 |- ( le ` K ) = ( le ` K )
3		eqid 2450	. . . . . . . . . 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 34746	. . . . . . . . 9 |- ( ( K e. V /\ W e. H ) -> dom I = { p e. ( Atoms ` K ) | -. p ( le ` K ) W } )
7	6	eleq2d 2513	. . . . . . . 8 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } ) )
8		breq1 4404	. . . . . . . . . 10 |- ( p = X -> ( p ( le ` K ) W <-> X ( le ` K ) W ) )
9	8	notbid 296	. . . . . . . . 9 |- ( p = X -> ( -. p ( le ` K ) W <-> -. X ( le ` K ) W ) )
10	9	elrab 3195	. . . . . . . 8 |- ( X e. { p e. ( Atoms ` K ) | -. p ( le ` K ) W } <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
11	7, 10	syl6bb 265	. . . . . . 7 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) ) )
12	11	biimpa 487	. . . . . 6 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( X e. ( Atoms ` K ) /\ -. X ( le ` K ) W ) )
13		eqid 2450	. . . . . . 7 |- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
14		eqid 2450	. . . . . . 7 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
15		eqid 2450	. . . . . . 7 |- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
16	2, 3, 4, 13, 14, 15, 5	dicval 34738	. . . . . 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 473	. . . . 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 4918	. . . 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 237	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
20	19	ex 436	. 2 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I -> Rel ( I ` X ) ) )
21		rel0 4957	. . 3 |- Rel (/)
22		ndmfv 5887	. . . 4 |- ( -. X e. dom I -> ( I ` X ) = (/) )
23	22	releqd 4918	. . 3 |- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
24	21, 23	mpbiri 237	. 2 |- ( -. X e. dom I -> Rel ( I ` X ) )
25	20, 24	pm2.61d1 163	1 |- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1443   e. wcel 1886   {crab 2740   (/)c0 3730   class class class wbr 4401   {copab 4459   dom cdm 4833   Rel wrel 4838   ` cfv 5581   iota_crio 6249   lecple 15190   occoc 15191   Atomscatm 32823   LHypclh 33543   LTrncltrn 33660   TEndoctendo 34313   DIsoCcdic 34734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-dic 34735

This theorem is referenced by: (None)