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Mathbox for Alexander van der Vekens |
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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 6610 | . . 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 47069 | . 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 2106 {csn 4631 dom cdm 5689 ↾ cres 5691 Fun wfun 6557 defAt wdfat 47066 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-in 3970 df-ss 3980 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-res 5701 df-fun 6565 df-dfat 47069 |
This theorem is referenced by: afv2elrn 47181 fnbrafv2b 47198 |
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