Theorem afv20defat 48128

Description: If the alternate function value at an argument is the empty set, the function is defined at this argument. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
afv20defat	⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)

Proof of Theorem afv20defat

Step	Hyp	Ref	Expression
1		ndfatafv2 48107	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwne0 5325	. . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅
3	2	neii 2959	. . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅
4		eqeq1 2766	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅))
5	3, 4	mtbiri 330	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ¬ (𝐹''''𝐴) = ∅)
6	1, 5	syl 18	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) = ∅)
7	6	con4i 115	1 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∅c0 4282  𝒫 cpw 4560  ∪ cuni 4870  ran crn 5660   defAt wdfat 48012  ''''cafv2 48104

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 2147  ax-9 2155  ax-ext 2734  ax-nul 5267

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-afv2 48105

This theorem is used by:  afv20fv0  48159