Theorem diafn 36323

Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)

Hypotheses

Ref	Expression
diafn.b	|- B = ( Base ` K )
diafn.l	|- .<_ = ( le ` K )
diafn.h	|- H = ( LHyp ` K )
diafn.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
diafn	|- ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )

Distinct variable groups:   x, .<_   x, B   x, K   x, W

Allowed substitution hints:   H( x)   I( x)   V( x)

Proof of Theorem diafn

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

Step	Hyp	Ref	Expression
1		fvex 6201	. . . 4 |- ( ( LTrn ` K ) ` W ) e. _V
2	1	rabex 4813	. . 3 |- { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } e. _V
3		eqid 2622	. . 3 |- ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) = ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } )
4	2, 3	fnmpti 6022	. 2 |- ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) Fn { x e. B | x .<_ W }
5		diafn.b	. . . 4 |- B = ( Base ` K )
6		diafn.l	. . . 4 |- .<_ = ( le ` K )
7		diafn.h	. . . 4 |- H = ( LHyp ` K )
8		eqid 2622	. . . 4 |- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
9		eqid 2622	. . . 4 |- ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
10		diafn.i	. . . 4 |- I = ( ( DIsoA ` K ) ` W )
11	5, 6, 7, 8, 9, 10	diafval 36320	. . 3 |- ( ( K e. V /\ W e. H ) -> I = ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) )
12	11	fneq1d 5981	. 2 |- ( ( K e. V /\ W e. H ) -> ( I Fn { x e. B | x .<_ W } <-> ( y e. { x e. B | x .<_ W } |-> { f e. ( ( LTrn ` K ) ` W ) | ( ( ( trL ` K ) ` W ) ` f ) .<_ y } ) Fn { x e. B | x .<_ W } ) )
13	4, 12	mpbiri 248	1 |- ( ( K e. V /\ W e. H ) -> I Fn { x e. B | x .<_ W } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888   Basecbs 15857   lecple 15948   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   DIsoAcdia 36317

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-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-pr 4906

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-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-disoa 36318

This theorem is referenced by:  diadm  36324  diaelrnN  36334  diaf11N  36338