Theorem funfni 6428

Description: Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)

Hypothesis

Ref	Expression
funfni.1	⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)

Assertion

Ref	Expression
funfni	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

Proof of Theorem funfni

Step	Hyp	Ref	Expression
1		fnfun 6423	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fndm 6425	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	eleq2d 2875	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
4	3	biimpar 481	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
5		funfni.1	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)
6	1, 4, 5	syl2an2r 684	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  dom cdm 5519  Fun wfun 6318   Fn wfn 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-cleq 2791  df-clel 2870  df-fn 6327

This theorem is referenced by:  fneu  6432  elpreima  6805  fnopfv  6820  fnfvelrn  6825  funressnfv  43635  fnafvelrn  43725  afvco2  43732  fnafv2elrn  43789  fnbrafv2b  43804