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Theorem stafval 15613
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 5525 . 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 15612 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
6 fvex 5539 . 2  |-  (  .* 
`  A )  e. 
_V
71, 5, 6fvmpt 5602 1  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   ` cfv 5255   Basecbs 13148   * rcstv 13210   * r fcstf 15608
This theorem is referenced by:  srngcl  15620  srngnvl  15621  srngadd  15622  srngmul  15623  srng1  15624  srng0  15625  issrngd  15626  iporthcom  16539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-staf 15610
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