Theorem funfnd 5154

Description: A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
funfnd.1	⊢ (𝜑 → Fun 𝐴)

Assertion

Ref	Expression
funfnd	⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Proof of Theorem funfnd

Step	Hyp	Ref	Expression
1		funfnd.1	. 2 ⊢ (𝜑 → Fun 𝐴)
2		funfn 5153	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 121	1 ⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Syntax hints:   → wi 4  dom cdm 4539  Fun wfun 5117   Fn wfn 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-gen 1425  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-cleq 2132  df-fn 5126

This theorem is referenced by:  ennnfonelemf1  11934  dvfgg  12829