Theorem afv20defat 43506

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 43485	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwne0 5250	. . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅
3	2	neii 3017	. . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅
4		eqeq1 2824	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅))
5	3, 4	mtbiri 329	. . 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 1536  ∅c0 4284  𝒫 cpw 4532  ∪ cuni 4831  ran crn 5549   defAt wdfat 43390  ''''cafv2 43482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-nul 5203

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-v 3493  df-dif 3932  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-pw 4534  df-afv2 43483

This theorem is referenced by:  afv20fv0  43537