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Theorem staffn 15939
Description: The functionalization is equal to the original function, if it is a function on the right base set. (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
staffn  |-  (  .*  Fn  B  ->  .xb  =  .*  )

Proof of Theorem staffn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dffn5 5774 . . 3  |-  (  .*  Fn  B  <->  .*  =  ( x  e.  B  |->  (  .*  `  x
) ) )
21biimpi 188 . 2  |-  (  .*  Fn  B  ->  .*  =  ( x  e.  B  |->  (  .*  `  x
) ) )
3 staffval.b . . 3  |-  B  =  ( Base `  R
)
4 staffval.i . . 3  |-  .*  =  ( * r `  R )
5 staffval.f . . 3  |-  .xb  =  ( * r f `
 R )
63, 4, 5staffval 15937 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
72, 6syl6reqr 2489 1  |-  (  .*  Fn  B  ->  .xb  =  .*  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. cmpt 4268    Fn wfn 5451   ` cfv 5456   Basecbs 13471   * rcstv 13533   * r fcstf 15933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-staf 15935
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