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Theorem staffn 19612
 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 218 . 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 19610 . 2 = (𝑥𝐵 ↦ ( 𝑥))
72, 6syl6reqr 2873 1 ( Fn 𝐵 = )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ↦ cmpt 5137   Fn wfn 6343  ‘cfv 6348  Basecbs 16475  *𝑟cstv 16559  *rfcstf 19606 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 19608 This theorem is referenced by: (None)
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