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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfatprc | Structured version Visualization version GIF version | ||
| Description: A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.) | 
| Ref | Expression | 
|---|---|
| dfatprc | ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | prcnel 3506 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹) | |
| 2 | 1 | orcd 873 | . 2 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | 
| 3 | ianor 983 | . . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
| 4 | df-dfat 47136 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 5 | 3, 4 | xchnxbir 333 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | 
| 6 | 2, 5 | sylibr 234 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 847 ∈ wcel 2107 Vcvv 3479 {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-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dfat 47136 | 
| This theorem is referenced by: (None) | 
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