Theorem fundmdfat 41983

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 6143	. . . 4 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim2i 611	. . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3	2	ancoms 451	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4		df-dfat 41973	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
5	3, 4	sylibr 226	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157  {csn 4368  dom cdm 5312   ↾ cres 5314  Fun wfun 6095   defAt wdfat 41970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-br 4844  df-opab 4906  df-rel 5319  df-cnv 5320  df-co 5321  df-res 5324  df-fun 6103  df-dfat 41973

This theorem is referenced by:  afv2elrn  42085  fnbrafv2b  42102