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Theorem staffn 15614
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 5568 . . 3  |-  (  .*  Fn  B  <->  .*  =  ( x  e.  B  |->  (  .*  `  x
) ) )
21biimpi 186 . 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 15612 . 2  |-  .xb  =  ( x  e.  B  |->  (  .*  `  x
) )
72, 6syl6reqr 2334 1  |-  (  .*  Fn  B  ->  .xb  =  .*  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. cmpt 4077    Fn wfn 5250   ` cfv 5255   Basecbs 13148   * rcstv 13210   * r fcstf 15608
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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