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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 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Proof of Theorem fundmdfat
StepHypRef Expression
1 funres 6143 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim2i 611 . . 3 ((𝐴 ∈ dom 𝐹 ∧ Fun 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
32ancoms 451 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4 df-dfat 41973 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
53, 4sylibr 226 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
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
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