Theorem funfni 6239

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 6235	. 2 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2		fndm 6237	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3	2	eleq2d 2845	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
4	3	biimpar 471	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ dom 𝐹)
5		funfni.1	. 2 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)
6	1, 4, 5	syl2an2r 675	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)

Syntax hints:   → wi 4   ∧ wa 386   ∈ wcel 2107  dom cdm 5357  Fun wfun 6131   Fn wfn 6132

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1824  df-cleq 2770  df-clel 2774  df-fn 6140

This theorem is referenced by:  fneu  6243  elpreima  6602  fnopfv  6617  fnfvelrn  6622  funressnfv  42117  fnafvelrn  42220  afvco2  42227  fnafv2elrn  42284  fnbrafv2b  42299