Theorem funfni 6661

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 6655	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fndm 6658	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	eleq2d 2811	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
4	3	biimpar 476	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
5		funfni.1	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)
6	1, 4, 5	syl2an2r 683	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

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