MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funfni Structured version   Visualization version   GIF version

Theorem funfni 6450
Description: Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
Hypothesis
Ref Expression
funfni.1 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
Assertion
Ref Expression
funfni ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)

Proof of Theorem funfni
StepHypRef Expression
1 fnfun 6446 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 6448 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2895 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
43biimpar 478 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
5 funfni.1 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
61, 4, 5syl2an2r 681 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  dom cdm 5548  Fun wfun 6342   Fn wfn 6343
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-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772  df-cleq 2811  df-clel 2890  df-fn 6351
This theorem is referenced by:  fneu  6454  elpreima  6820  fnopfv  6835  fnfvelrn  6840  funressnfv  43155  fnafvelrn  43245  afvco2  43252  fnafv2elrn  43309  fnbrafv2b  43324
  Copyright terms: Public domain W3C validator