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Theorem staffval 16910
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.
StepHypRef Expression
1 staffval.f . 2  |-  .xb  =  ( *rf `  R )
2 fveq2 5686 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2488 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5686 . . . . . . 7  |-  ( f  =  R  ->  (
*r `  f
)  =  ( *r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( *r `  R )
75, 6syl6eqr 2488 . . . . . 6  |-  ( f  =  R  ->  (
*r `  f
)  =  .*  )
87fveq1d 5688 . . . . 5  |-  ( f  =  R  ->  (
( *r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4365 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( *r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 16908 . . . 4  |-  *rf  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( *r `  f ) `
 x ) ) )
11 eqid 2438 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5707 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 11 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5861 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5696 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2508 . . . . 5  |-  B  e. 
_V
17 fvex 5696 . . . . . . . 8  |-  ( *r `  R )  e.  _V
186, 17eqeltri 2508 . . . . . . 7  |-  .*  e.  _V
1918rnex 6507 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4474 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 6373 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 6527 . . . . 5  |-  ( ( ( x  e.  B  |->  (  .*  `  x
) ) : B --> ( ran  .*  u.  { (/)
} )  /\  B  e.  _V  /\  ( ran 
.*  u.  { (/) } )  e.  _V )  -> 
( x  e.  B  |->  (  .*  `  x
) )  e.  _V )
2314, 16, 21, 22mp3an 1314 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5769 . . 3  |-  ( R  e.  _V  ->  (
*rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5680 . . . . 5  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  (/) )
26 mpt0 5533 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2488 . . . 4  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5680 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2482 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3029mpteq1d 4368 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3127, 30eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3224, 31pm2.61i 164 . 2  |-  ( *rf `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
331, 32eqtri 2458 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1369    e. wcel 1756   _Vcvv 2967    u. cun 3321   (/)c0 3632   {csn 3872    e. cmpt 4345   ran crn 4836   -->wf 5409   ` cfv 5413   Basecbs 14166   *rcstv 14232   *rfcstf 16906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-staf 16908
This theorem is referenced by:  stafval  16911  staffn  16912  issrngd  16924
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