Theorem funfnd 5411

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 5410	. 2 |- ( Fun A <-> A Fn dom A )
3	1, 2	sylib 122	1 |- ( ph -> A Fn dom A )

This proof depends on syntax axioms:   -> wi 4   dom cdm 4776   Fun wfun 5374   Fn wfn 5375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-gen 1502  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-cleq 2231  df-fn 5383

This theorem is used by:  fncofn  5896  mptsuppdifd  6499  funsssuppss  6502  suppcofn  6510  ccatalpha  11403  ennnfonelemf1  13367  dvfgg  15914  lpvtx  16525  uhgrvtxedgiedgb  16589  uhgr2edg  16652  ushgredgedg  16672  ushgredgedgloop  16674  subgruhgredgdm  16716  subuhgr  16718  subupgr  16719  subumgr  16720  subusgr  16721  vtxdfifiun  16743  trlsegvdegfi  16913  eupth2lem3lem2fi  16915  eupth2lem3lem6fi  16917  eupth2lem3lem4fi  16919  eupthvdres  16921