Mathbox for Alexander van der Vekens |
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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 3521 | . . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹) | |
2 | 1 | orcd 869 | . 2 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) |
3 | ianor 978 | . . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) | |
4 | df-dfat 43325 | . . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
5 | 3, 4 | xchnxbir 335 | . 2 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴}))) |
6 | 2, 5 | sylibr 236 | 1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∈ wcel 2113 Vcvv 3497 {csn 4570 dom cdm 5558 ↾ cres 5560 Fun wfun 6352 defAt wdfat 43322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1780 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-v 3499 df-dfat 43325 |
This theorem is referenced by: (None) |
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