Theorem fundmdfat 43322

Description: A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.)

Assertion

Ref	Expression
fundmdfat	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Proof of Theorem fundmdfat

Step	Hyp	Ref	Expression
1		funres 6391	. . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
2	1	anim1ci 617	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
3		df-dfat 43312	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4	2, 3	sylibr 236	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110  {csn 4560  dom cdm 5549   ↾ cres 5551  Fun wfun 6343   defAt wdfat 43309

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 2157  ax-12 2173  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 3496  df-in 3942  df-ss 3951  df-br 5059  df-opab 5121  df-rel 5556  df-cnv 5557  df-co 5558  df-res 5561  df-fun 6351  df-dfat 43312

This theorem is referenced by:  afv2elrn  43424  fnbrafv2b  43441