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Theorem staffval 17691
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 5848 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2513 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5848 . . . . . . 7  |-  ( f  =  R  ->  (
*r `  f
)  =  ( *r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( *r `  R )
75, 6syl6eqr 2513 . . . . . 6  |-  ( f  =  R  ->  (
*r `  f
)  =  .*  )
87fveq1d 5850 . . . . 5  |-  ( f  =  R  ->  (
( *r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4517 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( *r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 17689 . . . 4  |-  *rf  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( *r `  f ) `
 x ) ) )
11 eqid 2454 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5870 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 11 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 6030 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5858 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2538 . . . . 5  |-  B  e. 
_V
17 fvex 5858 . . . . . . . 8  |-  ( *r `  R )  e.  _V
186, 17eqeltri 2538 . . . . . . 7  |-  .*  e.  _V
1918rnex 6707 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4624 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 6571 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 6728 . . . . 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 1322 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5931 . . 3  |-  ( R  e.  _V  ->  (
*rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5842 . . . . 5  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  (/) )
26 mpt0 5690 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2513 . . . 4  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5842 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2507 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3029mpteq1d 4520 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3127, 30eqtr4d 2498 . . 3  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3224, 31pm2.61i 164 . 2  |-  ( *rf `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
331, 32eqtri 2483 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459   (/)c0 3783   {csn 4016    |-> cmpt 4497   ran crn 4989   -->wf 5566   ` cfv 5570   Basecbs 14716   *rcstv 14786   *rfcstf 17687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-staf 17689
This theorem is referenced by:  stafval  17692  staffn  17693  issrngd  17705
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