Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  staffn Structured version   Unicode version

Theorem staffn 17816
 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
staffval.i
staffval.f
Assertion
Ref Expression
staffn

Proof of Theorem staffn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffn5 5893 . . 3
21biimpi 194 . 2
3 staffval.b . . 3
4 staffval.i . . 3
5 staffval.f . . 3
63, 4, 5staffval 17814 . 2
72, 6syl6reqr 2462 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1405   cmpt 4452   wfn 5563  cfv 5568  cbs 14839  cstv 14909  cstf 17810 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-staf 17812 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator