Theorem fundmdfat 47894

Description: A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
fundmdfat	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Proof of Theorem fundmdfat

Step	Hyp	Ref	Expression
1		funres 6578	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim1ci 627	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47884	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	sylibr 237	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  {csn 4588  dom cdm 5660   ↾ cres 5662  Fun wfun 6530   defAt wdfat 47881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-ss 3921  df-br 5109  df-opab 5173  df-rel 5667  df-cnv 5668  df-co 5669  df-res 5672  df-fun 6538  df-dfat 47884

This theorem is used by:  afv2elrn  47996  fnbrafv2b  48013