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Theorem fundmdfat 47146
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 6607 . . 3 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1ci 616 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 47136 . 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 2107  {csn 4625  dom cdm 5684  cres 5686  Fun wfun 6554   defAt wdfat 47133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-in 3957  df-ss 3967  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-res 5696  df-fun 6562  df-dfat 47136
This theorem is referenced by:  afv2elrn  47248  fnbrafv2b  47265
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