Theorem dfatprc 44622

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 3455	. . 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 44611	. . 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 233	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐹 defAt 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 844   ∈ wcel 2106  Vcvv 3432  {csn 4561  dom cdm 5589   ↾ cres 5591  Fun wfun 6427   defAt wdfat 44608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dfat 44611

This theorem is referenced by: (None)