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

Theorem funfni 6609
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 6603 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 6606 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2824 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
43biimpar 479 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
5 funfni.1 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
61, 4, 5syl2an2r 684 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  dom cdm 5634  Fun wfun 6491   Fn wfn 6492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-cleq 2729  df-clel 2815  df-fn 6500
This theorem is referenced by:  fneu  6613  elpreima  7009  fnopfv  7027  fnfvelrn  7032  funressnfv  45284  fnafvelrn  45408  afvco2  45415  fnafv2elrn  45472  fnbrafv2b  45487
  Copyright terms: Public domain W3C validator