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Theorem staffn 19739
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 staffval.b . . 3 𝐵 = (Base‘𝑅)
2 staffval.i . . 3 = (*𝑟𝑅)
3 staffval.f . . 3 = (*rf𝑅)
41, 2, 3staffval 19737 . 2 = (𝑥𝐵 ↦ ( 𝑥))
5 dffn5 6728 . . 3 ( Fn 𝐵 = (𝑥𝐵 ↦ ( 𝑥)))
65biimpi 219 . 2 ( Fn 𝐵 = (𝑥𝐵 ↦ ( 𝑥)))
74, 6eqtr4id 2792 1 ( Fn 𝐵 = )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cmpt 5110   Fn wfn 6334  cfv 6339  Basecbs 16586  *𝑟cstv 16670  *rfcstf 19733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-staf 19735
This theorem is referenced by: (None)
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