Theorem dfatprc 43686

Description: A function is not defined at a proper class. (Contributed by AV, 1-Sep-2022.)

Assertion

Ref	Expression
dfatprc	⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Proof of Theorem dfatprc

Step	Hyp	Ref	Expression
1		prcnel 3465	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
2	1	orcd 870	. 2 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		ianor 979	. . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
4		df-dfat 43675	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
5	3, 4	xchnxbir 336	. 2 ⊢ (¬ 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
6	2, 5	sylibr 237	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∈ wcel 2111  Vcvv 3441  {csn 4525  dom cdm 5519   ↾ cres 5521  Fun wfun 6318   defAt wdfat 43672

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dfat 43675

This theorem is referenced by: (None)