Theorem afv2ndefb 47845

Description: Two ways to say that an alternate function value is not defined. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
afv2ndefb	⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem afv2ndefb

Step	Hyp	Ref	Expression
1		pwuninel 8267	. . 3 ⊢ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹
2		df-nel 3071	. . . 4 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
3		eleq1 2857	. . . . 5 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
4	3	notbid 321	. . . 4 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
5	2, 4	bitrid 286	. . 3 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ∪ ran 𝐹 ∈ ran 𝐹))
6	1, 5	mpbiri 261	. 2 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
7		funressndmafv2rn 47844	. . . . 5 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
8	7	con3i 155	. . . 4 ⊢ (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
9	2, 8	sylbi 220	. . 3 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
10		ndfatafv2 47832	. . 3 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
11	9, 10	syl 18	. 2 ⊢ ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝒫 ∪ ran 𝐹)
12	6, 11	impbii 212	1 ⊢ ((𝐹''''𝐴) = 𝒫 ∪ ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1567   ∈ wcel 2149   ∉ wnel 3070  𝒫 cpw 4564  ∪ cuni 4873  ran crn 5660   defAt wdfat 47737  ''''cafv2 47829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6490  df-fun 6536  df-dfat 47740  df-afv2 47830

This theorem is referenced by: (None)