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Theorem staffval 15705
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 5605 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2408 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5605 . . . . . . 7  |-  ( f  =  R  ->  (
* r `  f
)  =  ( * r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( * r `  R )
75, 6syl6eqr 2408 . . . . . 6  |-  ( f  =  R  ->  (
* r `  f
)  =  .*  )
87fveq1d 5607 . . . . 5  |-  ( f  =  R  ->  (
( * r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4177 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( * r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 15703 . . . 4  |-  * r f  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( * r `  f ) `
 x ) ) )
11 eqid 2358 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5630 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 10 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5763 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5619 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2428 . . . . 5  |-  B  e. 
_V
17 fvex 5619 . . . . . . . 8  |-  ( * r `  R )  e.  _V
186, 17eqeltri 2428 . . . . . . 7  |-  .*  e.  _V
1918rnex 5021 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4276 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 4597 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 5481 . . . . 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 5682 . . 3  |-  ( R  e.  _V  ->  (
* r f `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5599 . . . . 5  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  (/) )
26 mpt0 5450 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2408 . . . 4  |-  ( -.  R  e.  _V  ->  ( * r f `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5599 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2402 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
30 mpteq1 4179 . . . . 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 2393 . . 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 2378 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864    u. cun 3226   (/)c0 3531   {csn 3716    e. cmpt 4156   ran crn 4769   -->wf 5330   ` cfv 5334   Basecbs 13239   * rcstv 13301   * r fcstf 15701
This theorem is referenced by:  stafval  15706  staffn  15707  issrngd  15719
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-fv 5342  df-staf 15703
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