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Theorem staffval 15940
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  =  ( * r f `
 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  =  ( * r f `
 R )
2 fveq2 5731 . . . . . 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 5731 . . . . . . 7  |-  ( f  =  R  ->  (
* r `  f
)  =  ( * r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( * r `  R )
75, 6syl6eqr 2488 . . . . . 6  |-  ( f  =  R  ->  (
* r `  f
)  =  .*  )
87fveq1d 5733 . . . . 5  |-  ( f  =  R  ->  (
( * r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4290 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( * r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 15938 . . . 4  |-  * r f  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( * r `  f ) `
 x ) ) )
11 eqid 2438 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5756 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 11 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5895 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5745 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2508 . . . . 5  |-  B  e. 
_V
17 fvex 5745 . . . . . . . 8  |-  ( * r `  R )  e.  _V
186, 17eqeltri 2508 . . . . . . 7  |-  .*  e.  _V
1918rnex 5136 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4389 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 4710 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 5606 . . . . 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 1280 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5809 . . 3  |-  ( R  e.  _V  ->  (
* r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5725 . . . . 5  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  (/) )
26 mpt0 5575 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2488 . . . 4  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5725 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2482 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3029mpteq1d 4293 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3127, 30eqtr4d 2473 . . 3  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3224, 31pm2.61i 159 . 2  |-  ( * r f `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
331, 32eqtri 2458 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320   (/)c0 3630   {csn 3816    e. cmpt 4269   ran crn 4882   -->wf 5453   ` cfv 5457   Basecbs 13474   * rcstv 13536   * r fcstf 15936
This theorem is referenced by:  stafval  15941  staffn  15942  issrngd  15954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-staf 15938
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