Theorem funfni 5184

Description: Inference to convert a function and domain antecedent. (Contributed by set.mm contributors, 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 5182	. . 3 ⊢ (F Fn A → Fun F)
2	1	adantr 451	. 2 ⊢ ((F Fn A ∧ B ∈ A) → Fun F)
3		fndm 5183	. . . 4 ⊢ (F Fn A → dom F = A)
4	3	eleq2d 2420	. . 3 ⊢ (F Fn A → (B ∈ dom F ↔ B ∈ A))
5	4	biimpar 471	. 2 ⊢ ((F Fn A ∧ B ∈ A) → B ∈ dom F)
6		funfni.1	. 2 ⊢ ((Fun F ∧ B ∈ dom F) → φ)
7	2, 5, 6	syl2anc 642	1 ⊢ ((F Fn A ∧ B ∈ A) → φ)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∈ wcel 1710  dom cdm 4773  Fun wfun 4776   Fn wfn 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2346  df-clel 2349  df-fn 4791

This theorem is used by:  fneu  5188  elpreima  5408  fnopfv  5413  fnfvelrn  5415