Theorem funfni 2724

Description: Inference to convert a function and domain antecedent.

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		funfni.1	. 2 ⊢ ((Fun F ∧ B ∈ dom F) → φ)
2		fnfun 2721	. . 3 ⊢ (F Fn A → Fun F)
3	2	adantr 306	. 2 ⊢ ((F Fn A ∧ B ∈ A) → Fun F)
4		fndm 2723	. . . 4 ⊢ (F Fn A → dom F = A)
5	4	eleq2d 1156	. . 3 ⊢ (F Fn A → (B ∈ dom F ↔ B ∈ A))
6	5	biimpar 325	. 2 ⊢ ((F Fn A ∧ B ∈ A) → B ∈ dom F)
7	1, 3, 6	sylanc 361	1 ⊢ ((F Fn A ∧ B ∈ A) → φ)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fvco2 2866  fnopfv 2887  fnfvrn 2889  isomin 2937  isofrlem 2939

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-fn 2433

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