Theorem funfnd 5408

Description: A function is a function over its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypothesis

Ref	Expression
funfnd.1	⊢ (𝜑 → Fun 𝐴)

Assertion

Ref	Expression
funfnd	⊢ (𝜑 → 𝐴 Fn dom 𝐴)

Proof of Theorem funfnd

Step	Hyp	Ref	Expression
1		funfnd.1	. 2 ⊢ (𝜑 → Fun 𝐴)
2		funfn 5407	. 2 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
3	1, 2	sylib 122	1 ⊢ (𝜑 → 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  11396  ennnfonelemf1  13360  dvfgg  15841  lpvtx  16442  uhgrvtxedgiedgb  16506  uhgr2edg  16569  ushgredgedg  16589  ushgredgedgloop  16591  subgruhgredgdm  16633  subuhgr  16635  subupgr  16636  subumgr  16637  subusgr  16638  vtxdfifiun  16660  trlsegvdegfi  16830  eupth2lem3lem2fi  16832  eupth2lem3lem6fi  16834  eupth2lem3lem4fi  16836  eupthvdres  16838