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Theorem staffval 15890
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 5687 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2454 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5687 . . . . . . 7  |-  ( f  =  R  ->  (
* r `  f
)  =  ( * r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( * r `  R )
75, 6syl6eqr 2454 . . . . . 6  |-  ( f  =  R  ->  (
* r `  f
)  =  .*  )
87fveq1d 5689 . . . . 5  |-  ( f  =  R  ->  (
( * r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4247 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( * r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 15888 . . . 4  |-  * r f  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( * r `  f ) `
 x ) ) )
11 eqid 2404 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5712 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 11 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5851 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5701 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2474 . . . . 5  |-  B  e. 
_V
17 fvex 5701 . . . . . . . 8  |-  ( * r `  R )  e.  _V
186, 17eqeltri 2474 . . . . . . 7  |-  .*  e.  _V
1918rnex 5092 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4346 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 4666 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 5562 . . . . 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 1279 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5765 . . 3  |-  ( R  e.  _V  ->  (
* r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5681 . . . . 5  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  (/) )
26 mpt0 5531 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2454 . . . 4  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5681 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2448 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3029mpteq1d 4250 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3127, 30eqtr4d 2439 . . 3  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3224, 31pm2.61i 158 . 2  |-  ( * r f `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
331, 32eqtri 2424 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   {csn 3774    e. cmpt 4226   ran crn 4838   -->wf 5409   ` cfv 5413   Basecbs 13424   * rcstv 13486   * r fcstf 15886
This theorem is referenced by:  stafval  15891  staffn  15892  issrngd  15904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-staf 15888
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