Theorem funfnd 5355

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 5354	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Syntax hints:   → wi 4  dom cdm 4723  Fun wfun 5318   Fn wfn 5319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1495  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fn 5327

This theorem is referenced by:  fncofn  5827  ccatalpha  11183  ennnfonelemf1  13032  dvfgg  15405  lpvtx  15923  uhgrvtxedgiedgb  15987  uhgr2edg  16050  ushgredgedg  16070  ushgredgedgloop  16072  vtxdfifiun  16108