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Theorem funfni 6661
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 6655 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 6658 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2811 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹𝐵𝐴))
43biimpar 476 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
5 funfni.1 . 2 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝜑)
61, 4, 5syl2an2r 683 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2098  dom cdm 5678  Fun wfun 6543   Fn wfn 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-cleq 2717  df-clel 2802  df-fn 6552
This theorem is referenced by:  fneu  6665  elpreima  7066  fnopfv  7084  fnfvelrn  7089  funressnfv  46563  fnafvelrn  46687  afvco2  46694  fnafv2elrn  46751  fnbrafv2b  46766
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