Theorem dfatprc 47161

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 3462	. . 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 47150	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   ∈ wcel 2111  Vcvv 3436  {csn 4571  dom cdm 5611   ↾ cres 5613  Fun wfun 6470   defAt wdfat 47147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dfat 47150

This theorem is referenced by: (None)