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Theorem stafval 17048
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
stafval  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )

Proof of Theorem stafval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5792 . 2  |-  ( x  =  A  ->  (  .*  `  x )  =  (  .*  `  A
) )
2 staffval.b . . 3  |-  B  =  ( Base `  R
)
3 staffval.i . . 3  |-  .*  =  ( *r `  R )
4 staffval.f . . 3  |-  .xb  =  ( *rf `  R )
52, 3, 4staffval 17047 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
6 fvex 5802 . 2  |-  (  .* 
`  A )  e. 
_V
71, 5, 6fvmpt 5876 1  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   ` 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:  srngcl  17055  srngnvl  17056  srngadd  17057  srngmul  17058  srng1  17059  srng0  17060  issrngd  17061  iporthcom  18182
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