Theorem dibvalrel 36452

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

Hypotheses

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

Assertion

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

Proof of Theorem dibvalrel

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . 3 |- Rel ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } )
2		dibcl.h	. . . . . . . 8 |- H = ( LHyp ` K )
3		eqid 2622	. . . . . . . 8 |- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
4		dibcl.i	. . . . . . . 8 |- I = ( ( DIsoB ` K ) ` W )
5	2, 3, 4	dibdiadm 36444	. . . . . . 7 |- ( ( K e. V /\ W e. H ) -> dom I = dom ( ( DIsoA ` K ) ` W ) )
6	5	eleq2d 2687	. . . . . 6 |- ( ( K e. V /\ W e. H ) -> ( X e. dom I <-> X e. dom ( ( DIsoA ` K ) ` W ) ) )
7	6	biimpa 501	. . . . 5 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> X e. dom ( ( DIsoA ` K ) ` W ) )
8		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
9		eqid 2622	. . . . . 6 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
10		eqid 2622	. . . . . 6 |- ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
11	8, 2, 9, 10, 3, 4	dibval 36431	. . . . 5 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom ( ( DIsoA ` K ) ` W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
12	7, 11	syldan 487	. . . 4 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
13	12	releqd 5203	. . 3 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> ( Rel ( I ` X ) <-> Rel ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) )
14	1, 13	mpbiri 248	. 2 |- ( ( ( K e. V /\ W e. H ) /\ X e. dom I ) -> Rel ( I ` X ) )
15		rel0 5243	. . . 4 |- Rel (/)
16		ndmfv 6218	. . . . 5 |- ( -. X e. dom I -> ( I ` X ) = (/) )
17	16	releqd 5203	. . . 4 |- ( -. X e. dom I -> ( Rel ( I ` X ) <-> Rel (/) ) )
18	15, 17	mpbiri 248	. . 3 |- ( -. X e. dom I -> Rel ( I ` X ) )
19	18	adantl 482	. 2 |- ( ( ( K e. V /\ W e. H ) /\ -. X e. dom I ) -> Rel ( I ` X ) )
20	14, 19	pm2.61dan 832	1 |- ( ( K e. V /\ W e. H ) -> Rel ( I ` X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   {csn 4177   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   |` cres 5116   Rel wrel 5119   ` cfv 5888   Basecbs 15857   LHypclh 35270   LTrncltrn 35387   DIsoAcdia 36317   DIsoBcdib 36427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-dib 36428

This theorem is referenced by:  dibglbN  36455  dib2dim  36532  dih2dimbALTN  36534