Theorem afv20defat 47702

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 47681	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
2		pwne0 5292	. . . . 5 ⊢ 𝒫 ∪ ran 𝐹 ≠ ∅
3	2	neii 2937	. . . 4 ⊢ ¬ 𝒫 ∪ ran 𝐹 = ∅
4		eqeq1 2744	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) = ∅ ↔ 𝒫 ∪ ran 𝐹 = ∅))
5	3, 4	mtbiri 328	. . 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 1547  ∅c0 4268  𝒫 cpw 4536  ∪ cuni 4845  ran crn 5626   defAt wdfat 47586  ''''cafv2 47678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-nul 5235

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-v 3434  df-dif 3893  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-afv2 47679

This theorem is referenced by:  afv20fv0  47733