Theorem funfnd 5411

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 5410	. 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 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  11404  ennnfonelemf1  13368  dvfgg  15926  lpvtx  16537  uhgrvtxedgiedgb  16601  uhgr2edg  16664  ushgredgedg  16684  ushgredgedgloop  16686  subgruhgredgdm  16728  subuhgr  16730  subupgr  16731  subumgr  16732  subusgr  16733  vtxdfifiun  16755  trlsegvdegfi  16925  eupth2lem3lem2fi  16927  eupth2lem3lem6fi  16929  eupth2lem3lem4fi  16931  eupthvdres  16933