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  11397  ennnfonelemf1  13361  dvfgg  15880  lpvtx  16491  uhgrvtxedgiedgb  16555  uhgr2edg  16618  ushgredgedg  16638  ushgredgedgloop  16640  subgruhgredgdm  16682  subuhgr  16684  subupgr  16685  subumgr  16686  subusgr  16687  vtxdfifiun  16709  trlsegvdegfi  16879  eupth2lem3lem2fi  16881  eupth2lem3lem6fi  16883  eupth2lem3lem4fi  16885  eupthvdres  16887