Theorem staffval 18847

Description: The functionalization of the involution component of a structure. (Contributed by Mario Carneiro, 6-Oct-2015.)

Hypotheses

Ref	Expression
staffval.b	|- B = ( Base ` R )
staffval.i	|- .* = ( *r ` R )
staffval.f	|- .xb = ( *rf ` R )

Assertion

Ref	Expression
staffval	|- .xb = ( x e. B |-> ( .* ` x ) )

Distinct variable groups:   x, B   x, .*   x, R

Allowed substitution hint:   .xb ( x)

Proof of Theorem staffval

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		staffval.f	. 2 |- .xb = ( *rf ` R )
2		fveq2 6191	. . . . . 6 |- ( f = R -> ( Base ` f ) = ( Base ` R ) )
3		staffval.b	. . . . . 6 |- B = ( Base ` R )
4	2, 3	syl6eqr 2674	. . . . 5 |- ( f = R -> ( Base ` f ) = B )
5		fveq2 6191	. . . . . . 7 |- ( f = R -> ( *r ` f ) = ( *r ` R ) )
6		staffval.i	. . . . . . 7 |- .* = ( *r ` R )
7	5, 6	syl6eqr 2674	. . . . . 6 |- ( f = R -> ( *r ` f ) = .* )
8	7	fveq1d 6193	. . . . 5 |- ( f = R -> ( ( *r ` f ) ` x ) = ( .* ` x ) )
9	4, 8	mpteq12dv 4733	. . . 4 |- ( f = R -> ( x e. ( Base ` f ) |-> ( ( *r ` f ) ` x ) ) = ( x e. B |-> ( .* ` x ) ) )
10		df-staf 18845	. . . 4 |- *rf = ( f e. _V |-> ( x e. ( Base ` f ) |-> ( ( *r ` f ) ` x ) ) )
11		eqid 2622	. . . . . 6 |- ( x e. B |-> ( .* ` x ) ) = ( x e. B |-> ( .* ` x ) )
12		fvrn0 6216	. . . . . . 7 |- ( .* ` x ) e. ( ran .* u. { (/) } )
13	12	a1i 11	. . . . . 6 |- ( x e. B -> ( .* ` x ) e. ( ran .* u. { (/) } ) )
14	11, 13	fmpti 6383	. . . . 5 |- ( x e. B |-> ( .* ` x ) ) : B --> ( ran .* u. { (/) } )
15		fvex 6201	. . . . . 6 |- ( Base ` R ) e. _V
16	3, 15	eqeltri 2697	. . . . 5 |- B e. _V
17		fvex 6201	. . . . . . . 8 |- ( *r ` R ) e. _V
18	6, 17	eqeltri 2697	. . . . . . 7 |- .* e. _V
19	18	rnex 7100	. . . . . 6 |- ran .* e. _V
20		p0ex 4853	. . . . . 6 |- { (/) } e. _V
21	19, 20	unex 6956	. . . . 5 |- ( ran .* u. { (/) } ) e. _V
22		fex2 7121	. . . . 5 |- ( ( ( x e. B |-> ( .* ` x ) ) : B --> ( ran .* u. { (/) } ) /\ B e. _V /\ ( ran .* u. { (/) } ) e. _V ) -> ( x e. B |-> ( .* ` x ) ) e. _V )
23	14, 16, 21, 22	mp3an 1424	. . . 4 |- ( x e. B |-> ( .* ` x ) ) e. _V
24	9, 10, 23	fvmpt 6282	. . 3 |- ( R e. _V -> ( *rf ` R ) = ( x e. B |-> ( .* ` x ) ) )
25		fvprc 6185	. . . . 5 |- ( -. R e. _V -> ( *rf ` R ) = (/) )
26		mpt0 6021	. . . . 5 |- ( x e. (/) |-> ( .* ` x ) ) = (/)
27	25, 26	syl6eqr 2674	. . . 4 |- ( -. R e. _V -> ( *rf ` R ) = ( x e. (/) |-> ( .* ` x ) ) )
28		fvprc 6185	. . . . . 6 |- ( -. R e. _V -> ( Base ` R ) = (/) )
29	3, 28	syl5eq 2668	. . . . 5 |- ( -. R e. _V -> B = (/) )
30	29	mpteq1d 4738	. . . 4 |- ( -. R e. _V -> ( x e. B |-> ( .* ` x ) ) = ( x e. (/) |-> ( .* ` x ) ) )
31	27, 30	eqtr4d 2659	. . 3 |- ( -. R e. _V -> ( *rf ` R ) = ( x e. B |-> ( .* ` x ) ) )
32	24, 31	pm2.61i 176	. 2 |- ( *rf ` R ) = ( x e. B |-> ( .* ` x ) )
33	1, 32	eqtri 2644	1 |- .xb = ( x e. B |-> ( .* ` x ) )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   (/)c0 3915   {csn 4177   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888   Basecbs 15857   *rcstv 15943   *rfcstf 18843

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-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-rab 2921  df-v 3202  df-sbc 3436  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-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-fv 5896  df-staf 18845

This theorem is referenced by:  stafval  18848  staffn  18849  issrngd  18861