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  11402  ennnfonelemf1  13366  dvfgg  15931  lpvtx  16542  uhgrvtxedgiedgb  16606  uhgr2edg  16669  ushgredgedg  16689  ushgredgedgloop  16691  subgruhgredgdm  16733  subuhgr  16735  subupgr  16736  subumgr  16737  subusgr  16738  vtxdfifiun  16760  trlsegvdegfi  16930  eupth2lem3lem2fi  16932  eupth2lem3lem6fi  16934  eupth2lem3lem4fi  16936  eupthvdres  16938