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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 6563 | . . 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 45504 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
4 | 2, 3 | sylibr 233 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 {csn 4606 dom cdm 5653 ↾ cres 5655 Fun wfun 6510 defAt wdfat 45501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3461 df-in 3935 df-ss 3945 df-br 5126 df-opab 5188 df-rel 5660 df-cnv 5661 df-co 5662 df-res 5665 df-fun 6518 df-dfat 45504 |
This theorem is referenced by: afv2elrn 45616 fnbrafv2b 45633 |
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