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Theorem staffn 17281
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  =  ( *rf `  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 5911 . . 3  |-  (  .*  Fn  B  <->  .*  =  ( x  e.  B  |->  (  .*  `  x
) ) )
21biimpi 194 . 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  =  ( *rf `  R )
63, 4, 5staffval 17279 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
72, 6syl6reqr 2527 1  |-  (  .*  Fn  B  ->  .xb  =  .*  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    |-> cmpt 4505    Fn wfn 5581   ` cfv 5586   Basecbs 14486   *rcstv 14553   *rfcstf 17275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-staf 17277
This theorem is referenced by: (None)
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