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Theorem staffval 15612
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 5525 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2333 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5525 . . . . . . 7  |-  ( f  =  R  ->  (
* r `  f
)  =  ( * r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( * r `  R )
75, 6syl6eqr 2333 . . . . . 6  |-  ( f  =  R  ->  (
* r `  f
)  =  .*  )
87fveq1d 5527 . . . . 5  |-  ( f  =  R  ->  (
( * r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4098 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( * r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 15610 . . . 4  |-  * r f  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( * r `  f ) `
 x ) ) )
11 eqid 2283 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5550 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 10 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5683 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5539 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2353 . . . . 5  |-  B  e. 
_V
17 fvex 5539 . . . . . . . 8  |-  ( * r `  R )  e.  _V
186, 17eqeltri 2353 . . . . . . 7  |-  .*  e.  _V
1918rnex 4942 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4197 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 4518 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 5401 . . . . 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 1277 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5602 . . 3  |-  ( R  e.  _V  ->  (
* r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5519 . . . . 5  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  (/) )
26 mpt0 5371 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2333 . . . 4  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5519 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2327 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
30 mpteq1 4100 . . . . 5  |-  ( B  =  (/)  ->  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3129, 30syl 15 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3227, 31eqtr4d 2318 . . 3  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3324, 32pm2.61i 156 . 2  |-  ( * r f `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
341, 33eqtri 2303 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640    e. cmpt 4077   ran crn 4690   -->wf 5251   ` cfv 5255   Basecbs 13148   * rcstv 13210   * r fcstf 15608
This theorem is referenced by:  stafval  15613  staffn  15614  issrngd  15626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-staf 15610
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