Theorem funfni 6621

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 6615	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fndm 6618	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	eleq2d 2847	. . 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 695	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  dom cdm 5643  Fun wfun 6509   Fn wfn 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1799  df-cleq 2753  df-clel 2836  df-fn 6518

This theorem is referenced by:  fneu  6625  elpreima  7033  fnopfv  7050  fnfvelrn  7055  funressnfv  47597  fnafvelrn  47723  afvco2  47730  fnafv2elrn  47787  fnbrafv2b  47802