Theorem fundmdfat 47120

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 6560	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim1ci 616	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47110	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	sylibr 234	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {csn 4591  dom cdm 5640   ↾ cres 5642  Fun wfun 6507   defAt wdfat 47107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-in 3923  df-ss 3933  df-br 5110  df-opab 5172  df-rel 5647  df-cnv 5648  df-co 5649  df-res 5652  df-fun 6515  df-dfat 47110

This theorem is referenced by:  afv2elrn  47222  fnbrafv2b  47239