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Theorem funfni 6532
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 6526 . 2 (𝐹 Fn 𝐴 → Fun 𝐹)
2 fndm 6529 . . . 4 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
32eleq2d 2824 . . 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 682 1 ((𝐹 Fn 𝐴𝐵𝐴) → 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  dom cdm 5585  Fun wfun 6421   Fn wfn 6422
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-clel 2816  df-fn 6430
This theorem is referenced by:  fneu  6536  elpreima  6928  fnopfv  6946  fnfvelrn  6951  funressnfv  44493  fnafvelrn  44617  afvco2  44624  fnafv2elrn  44681  fnbrafv2b  44696
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