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Theorem fundmdfat 43321
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 6392 . . 3 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1ci 617 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 43311 . 2 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3sylibr 236 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110  {csn 4561  dom cdm 5550  cres 5552  Fun wfun 6344   defAt wdfat 43308
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-in 3943  df-ss 3952  df-br 5060  df-opab 5122  df-rel 5557  df-cnv 5558  df-co 5559  df-res 5562  df-fun 6352  df-dfat 43311
This theorem is referenced by:  afv2elrn  43423  fnbrafv2b  43440
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