Theorem dibfna 36443

Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)

Hypotheses

Ref	Expression
dibfna.h	|- H = ( LHyp ` K )
dibfna.j	|- J = ( ( DIsoA ` K ) ` W )
dibfna.i	|- I = ( ( DIsoB ` K ) ` W )

Assertion

Ref	Expression
dibfna	|- ( ( K e. V /\ W e. H ) -> I Fn dom J )

Proof of Theorem dibfna

Dummy variables y f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( J ` y ) e. _V
2		snex 4908	. . . 4 |- { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } e. _V
3	1, 2	xpex 6962	. . 3 |- ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) e. _V
4		eqid 2622	. . 3 |- ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) = ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) )
5	3, 4	fnmpti 6022	. 2 |- ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) Fn dom J
6		eqid 2622	. . . 4 |- ( Base ` K ) = ( Base ` K )
7		dibfna.h	. . . 4 |- H = ( LHyp ` K )
8		eqid 2622	. . . 4 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
9		eqid 2622	. . . 4 |- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) )
10		dibfna.j	. . . 4 |- J = ( ( DIsoA ` K ) ` W )
11		dibfna.i	. . . 4 |- I = ( ( DIsoB ` K ) ` W )
12	6, 7, 8, 9, 10, 11	dibfval 36430	. . 3 |- ( ( K e. V /\ W e. H ) -> I = ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) )
13	12	fneq1d 5981	. 2 |- ( ( K e. V /\ W e. H ) -> ( I Fn dom J <-> ( y e. dom J |-> ( ( J ` y ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` ( Base ` K ) ) ) } ) ) Fn dom J ) )
14	5, 13	mpbiri 248	1 |- ( ( K e. V /\ W e. H ) -> I Fn dom J )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` 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:  dibdiadm  36444  dibfnN  36445  dibclN  36451