Theorem afv20defat 47514

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 47493	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwne0 5303	. . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅
3	2	neii 2935	. . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅
4		eqeq1 2741	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅))
5	3, 4	mtbiri 327	. . 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 1542  ∅c0 4286  𝒫 cpw 4555  ∪ cuni 4864  ran crn 5626   defAt wdfat 47398  ''''cafv2 47490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5252

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-v 3443  df-dif 3905  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-afv2 47491

This theorem is referenced by:  afv20fv0  47545