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Theorem staffval 17047
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 5792 . . . . . 6  |-  ( f  =  R  ->  ( Base `  f )  =  ( Base `  R
) )
3 staffval.b . . . . . 6  |-  B  =  ( Base `  R
)
42, 3syl6eqr 2510 . . . . 5  |-  ( f  =  R  ->  ( Base `  f )  =  B )
5 fveq2 5792 . . . . . . 7  |-  ( f  =  R  ->  (
*r `  f
)  =  ( *r `  R ) )
6 staffval.i . . . . . . 7  |-  .*  =  ( *r `  R )
75, 6syl6eqr 2510 . . . . . 6  |-  ( f  =  R  ->  (
*r `  f
)  =  .*  )
87fveq1d 5794 . . . . 5  |-  ( f  =  R  ->  (
( *r `  f ) `  x
)  =  (  .* 
`  x ) )
94, 8mpteq12dv 4471 . . . 4  |-  ( f  =  R  ->  (
x  e.  ( Base `  f )  |->  ( ( *r `  f
) `  x )
)  =  ( x  e.  B  |->  (  .* 
`  x ) ) )
10 df-staf 17045 . . . 4  |-  *rf  =  ( f  e.  _V  |->  ( x  e.  ( Base `  f
)  |->  ( ( *r `  f ) `
 x ) ) )
11 eqid 2451 . . . . . 6  |-  ( x  e.  B  |->  (  .* 
`  x ) )  =  ( x  e.  B  |->  (  .*  `  x ) )
12 fvrn0 5814 . . . . . . 7  |-  (  .* 
`  x )  e.  ( ran  .*  u.  {
(/) } )
1312a1i 11 . . . . . 6  |-  ( x  e.  B  ->  (  .*  `  x )  e.  ( ran  .*  u.  {
(/) } ) )
1411, 13fmpti 5968 . . . . 5  |-  ( x  e.  B  |->  (  .* 
`  x ) ) : B --> ( ran 
.*  u.  { (/) } )
15 fvex 5802 . . . . . 6  |-  ( Base `  R )  e.  _V
163, 15eqeltri 2535 . . . . 5  |-  B  e. 
_V
17 fvex 5802 . . . . . . . 8  |-  ( *r `  R )  e.  _V
186, 17eqeltri 2535 . . . . . . 7  |-  .*  e.  _V
1918rnex 6615 . . . . . 6  |-  ran  .*  e.  _V
20 p0ex 4580 . . . . . 6  |-  { (/) }  e.  _V
2119, 20unex 6481 . . . . 5  |-  ( ran 
.*  u.  { (/) } )  e.  _V
22 fex2 6635 . . . . 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 1315 . . . 4  |-  ( x  e.  B  |->  (  .* 
`  x ) )  e.  _V
249, 10, 23fvmpt 5876 . . 3  |-  ( R  e.  _V  ->  (
*rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
25 fvprc 5786 . . . . 5  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  (/) )
26 mpt0 5639 . . . . 5  |-  ( x  e.  (/)  |->  (  .*  `  x ) )  =  (/)
2725, 26syl6eqr 2510 . . . 4  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  (/)  |->  (  .* 
`  x ) ) )
28 fvprc 5786 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
293, 28syl5eq 2504 . . . . 5  |-  ( -.  R  e.  _V  ->  B  =  (/) )
3029mpteq1d 4474 . . . 4  |-  ( -.  R  e.  _V  ->  ( x  e.  B  |->  (  .*  `  x ) )  =  ( x  e.  (/)  |->  (  .*  `  x ) ) )
3127, 30eqtr4d 2495 . . 3  |-  ( -.  R  e.  _V  ->  ( *rf `  R )  =  ( x  e.  B  |->  (  .*  `  x ) ) )
3224, 31pm2.61i 164 . 2  |-  ( *rf `  R
)  =  ( x  e.  B  |->  (  .* 
`  x ) )
331, 32eqtri 2480 1  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1370    e. wcel 1758   _Vcvv 3071    u. cun 3427   (/)c0 3738   {csn 3978    |-> cmpt 4451   ran crn 4942   -->wf 5515   ` cfv 5519   Basecbs 14285   *rcstv 14351   *rfcstf 17043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-fv 5527  df-staf 17045
This theorem is referenced by:  stafval  17048  staffn  17049  issrngd  17061
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