Theorem dfatprc 47895

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 3480	. . 3 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ dom 𝐹)
2	1	orcd 886	. 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 47884	. . 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 400   ∨ wo 860   ∈ wcel 2143  Vcvv 3455  {csn 4589  dom cdm 5661   ↾ cres 5663  Fun wfun 6530   defAt wdfat 47881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dfat 47884

This theorem is used by: (None)