Theorem dfatprc 47945

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 3482	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
2	1	orcd 887	. 2 ⊢ (¬ 𝐴 ∈ V → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3		ianor 997	. . 3 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
4		df-dfat 47934	. . 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)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∈ wcel 2146  Vcvv 3457  {csn 4591  dom cdm 5663   ↾ cres 5665  Fun wfun 6535   defAt wdfat 47931

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-dfat 47934

This theorem is used by: (None)