Theorem fundmdfat 47899

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 6582	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim1ci 628	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 47889	. 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 401   ∈ wcel 2146  {csn 4591  dom cdm 5663   ↾ cres 5665  Fun wfun 6534   defAt wdfat 47886

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-in 3913  df-ss 3923  df-br 5112  df-opab 5176  df-rel 5670  df-cnv 5671  df-co 5672  df-res 5675  df-fun 6542  df-dfat 47889

This theorem is used by:  afv2elrn  48001  fnbrafv2b  48018