Theorem funfnd 6062

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 6061	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 208	1 ⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Syntax hints:   → wi 4  dom cdm 5249  Fun wfun 6025   Fn wfn 6026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1853  df-cleq 2764  df-fn 6034

This theorem is referenced by:  wfrlem4  7570  uhgrvtxedgiedgb  26252  ushgredgedgloop  26345  upgrres  26421  umgrres  26422  funimaeq  39979  limsupresxr  40516  liminfresxr  40517