Theorem fundmdfat 47684

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 6558	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim1ci 625	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47674	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	sylibr 236	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2141  {csn 4579  dom cdm 5643   ↾ cres 5645  Fun wfun 6510   defAt wdfat 47671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-in 3909  df-ss 3919  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-res 5655  df-fun 6518  df-dfat 47674

This theorem is referenced by:  afv2elrn  47786  fnbrafv2b  47803