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Theorem stafval 15899
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 5695 . 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 15898 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
6 fvex 5709 . 2  |-  (  .* 
`  A )  e. 
_V
71, 5, 6fvmpt 5773 1  |-  ( A  e.  B  ->  (  .xb  `  A )  =  (  .*  `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   ` cfv 5421   Basecbs 13432   * rcstv 13494   * r fcstf 15894
This theorem is referenced by:  srngcl  15906  srngnvl  15907  srngadd  15908  srngmul  15909  srng1  15910  srng0  15911  issrngd  15912  iporthcom  16829
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-staf 15896
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