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Theorem stafval 15941
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
stafval  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )

Proof of Theorem stafval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5731 . 2  |-  ( x  =  A  ->  (  .*  `  x )  =  (  .*  `  A
) )
2 staffval.b . . 3  |-  B  =  ( Base `  R
)
3 staffval.i . . 3  |-  .*  =  ( * r `  R )
4 staffval.f . . 3  |-  .xb  =  ( * r f `
 R )
52, 3, 4staffval 15940 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
6 fvex 5745 . 2  |-  (  .* 
`  A )  e. 
_V
71, 5, 6fvmpt 5809 1  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   ` cfv 5457   Basecbs 13474   * rcstv 13536   * r fcstf 15936
This theorem is referenced by:  srngcl  15948  srngnvl  15949  srngadd  15950  srngmul  15951  srng1  15952  srng0  15953  issrngd  15954  iporthcom  16871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-staf 15938
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