Theorem funfnd 5919

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 5918	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 208	1 ⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Syntax hints:   → wi 4  dom cdm 5114  Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-fn 5891

This theorem is referenced by:  upgrres  26198  umgrres  26199  funimaeq  39461  limsupresxr  39998  liminfresxr  39999