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Theorem funfni 5905
 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 5902 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 5904 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2673 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
43biimpar 501 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
5 funfni.1 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
61, 4, 5syl2an2r 872 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  dom cdm 5038  Fun wfun 5798   Fn wfn 5799 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-cleq 2603  df-clel 2606  df-fn 5807 This theorem is referenced by:  fneu  5909  elpreima  6245  fnopfv  6259  fnfvelrn  6264  funressnfv  39857  fnafvelrn  39898  afvco2  39905
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