Theorem afv20defat 44611

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 44590	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwne0 5274	. . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅
3	2	neii 2944	. . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅
4		eqeq1 2742	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅))
5	3, 4	mtbiri 326	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ¬ (𝐹''''𝐴) = ∅)
6	1, 5	syl 17	. 2 ⊢ (¬ 𝐹 defAt 𝐴 → ¬ (𝐹''''𝐴) = ∅)
7	6	con4i 114	1 ⊢ ((𝐹''''𝐴) = ∅ → 𝐹 defAt 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836  ran crn 5581   defAt wdfat 44495  ''''cafv2 44587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-afv2 44588

This theorem is referenced by:  afv20fv0  44642