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Theorem funfni 6239
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 6235 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 6237 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2845 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
43biimpar 471 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
5 funfni.1 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
61, 4, 5syl2an2r 675 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wcel 2107  dom cdm 5357  Fun wfun 6131   Fn wfn 6132
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-cleq 2770  df-clel 2774  df-fn 6140
This theorem is referenced by:  fneu  6243  elpreima  6602  fnopfv  6617  fnfvelrn  6622  funressnfv  42117  fnafvelrn  42220  afvco2  42227  fnafv2elrn  42284  fnbrafv2b  42299
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