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  11397  ennnfonelemf1  13361  dvfgg  15880  lpvtx  16486  uhgrvtxedgiedgb  16550  uhgr2edg  16613  ushgredgedg  16633  ushgredgedgloop  16635  subgruhgredgdm  16677  subuhgr  16679  subupgr  16680  subumgr  16681  subusgr  16682  vtxdfifiun  16704  trlsegvdegfi  16874  eupth2lem3lem2fi  16876  eupth2lem3lem6fi  16878  eupth2lem3lem4fi  16880  eupthvdres  16882