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  11381  ennnfonelemf1  13309  dvfgg  15789  lpvtx  16320  uhgrvtxedgiedgb  16384  uhgr2edg  16447  ushgredgedg  16467  ushgredgedgloop  16469  subgruhgredgdm  16511  subuhgr  16513  subupgr  16514  subumgr  16515  subusgr  16516  vtxdfifiun  16538  trlsegvdegfi  16708  eupth2lem3lem2fi  16710  eupth2lem3lem6fi  16712  eupth2lem3lem4fi  16714  eupthvdres  16716