Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fnafvelrn Structured version   Visualization version   GIF version

Theorem fnafvelrn 47181
Description: A function's value belongs to its range, analogous to fnfvelrn 7100. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fnafvelrn ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)

Proof of Theorem fnafvelrn
StepHypRef Expression
1 afvelrn 47180 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐹'''𝐵) ∈ ran 𝐹)
21funfni 6674 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  ran crn 5686   Fn wfn 6556  '''cafv 47129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132
This theorem is referenced by:  fafvelcdm  47182
  Copyright terms: Public domain W3C validator