Theorem funfnd 5408

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 5407	. 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 4774   Fun wfun 5371   Fn wfn 5372

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 5380

This theorem is used by:  fncofn  5893  mptsuppdifd  6495  funsssuppss  6498  suppcofn  6506  ccatalpha  11395  ennnfonelemf1  13358  dvfgg  15838  lpvtx  16418  uhgrvtxedgiedgb  16482  uhgr2edg  16545  ushgredgedg  16565  ushgredgedgloop  16567  subgruhgredgdm  16609  subuhgr  16611  subupgr  16612  subumgr  16613  subusgr  16614  vtxdfifiun  16636  trlsegvdegfi  16806  eupth2lem3lem2fi  16808  eupth2lem3lem6fi  16810  eupth2lem3lem4fi  16812  eupthvdres  16814