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Theorem fundmdfat 45514
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 6563 . . 3 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1ci 616 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 45504 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3sylibr 233 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  {csn 4606  dom cdm 5653  cres 5655  Fun wfun 6510   defAt wdfat 45501
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3461  df-in 3935  df-ss 3945  df-br 5126  df-opab 5188  df-rel 5660  df-cnv 5661  df-co 5662  df-res 5665  df-fun 6518  df-dfat 45504
This theorem is referenced by:  afv2elrn  45616  fnbrafv2b  45633
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