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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fundmdfat | Structured version Visualization version GIF version | ||
| Description: A function is defined at any element of its domain. (Contributed by AV, 2-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| fundmdfat | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funres 6607 | . . 3 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ {𝐴})) | |
| 2 | 1 | anim1ci 616 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | 
| 3 | df-dfat 47136 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 4 | 2, 3 | sylibr 234 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 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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