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Theorem fundmdfat 47079
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 6610 . . 3 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1ci 616 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 47069 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3sylibr 234 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  {csn 4631  dom cdm 5689  cres 5691  Fun wfun 6557   defAt wdfat 47066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-in 3970  df-ss 3980  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-res 5701  df-fun 6565  df-dfat 47069
This theorem is referenced by:  afv2elrn  47181  fnbrafv2b  47198
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