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Theorem fundmdfat 44053
 Description: A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
fundmdfat ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Proof of Theorem fundmdfat
StepHypRef Expression
1 funres 6377 . . 3 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1ci 618 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 44043 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3sylibr 237 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  {csn 4522  dom cdm 5524   ↾ cres 5526  Fun wfun 6329   defAt wdfat 44040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-br 5033  df-opab 5095  df-rel 5531  df-cnv 5532  df-co 5533  df-res 5536  df-fun 6337  df-dfat 44043 This theorem is referenced by:  afv2elrn  44155  fnbrafv2b  44172
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