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Theorem staffn 19549
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 = (*𝑟𝑅)
staffval.f = (*rf𝑅)
Assertion
Ref Expression
staffn ( Fn 𝐵 = )

Proof of Theorem staffn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dffn5 6717 . . 3 ( Fn 𝐵 = (𝑥𝐵 ↦ ( 𝑥)))
21biimpi 217 . 2 ( Fn 𝐵 = (𝑥𝐵 ↦ ( 𝑥)))
3 staffval.b . . 3 𝐵 = (Base‘𝑅)
4 staffval.i . . 3 = (*𝑟𝑅)
5 staffval.f . . 3 = (*rf𝑅)
63, 4, 5staffval 19547 . 2 = (𝑥𝐵 ↦ ( 𝑥))
72, 6syl6reqr 2872 1 ( Fn 𝐵 = )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  cmpt 5137   Fn wfn 6343  cfv 6348  Basecbs 16471  *𝑟cstv 16555  *rfcstf 19543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-staf 19545
This theorem is referenced by: (None)
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