Theorem funfnd 5357

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

Hypothesis

Ref	Expression
funfnd.1	|- ( ph -> Fun A )

Assertion

Ref	Expression
funfnd	|- ( ph -> A Fn dom A )

Proof of Theorem funfnd

Step	Hyp	Ref	Expression
1		funfnd.1	. 2 |- ( ph -> Fun A )
2		funfn 5356	. 2 |- ( Fun A <-> A Fn dom A )
3	1, 2	sylib 122	1 |- ( ph -> A Fn dom A )

Syntax hints:   -> wi 4   dom cdm 4725   Fun wfun 5320   Fn wfn 5321

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 1497  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-fn 5329

This theorem is referenced by:  fncofn  5831  ccatalpha  11189  ennnfonelemf1  13038  dvfgg  15411  lpvtx  15929  uhgrvtxedgiedgb  15993  uhgr2edg  16056  ushgredgedg  16076  ushgredgedgloop  16078  subgruhgredgdm  16120  subuhgr  16122  subupgr  16123  subumgr  16124  subusgr  16125  vtxdfifiun  16147