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Theorem afv2ndefb 46667
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
StepHypRef Expression
1 pwuninel 8279 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
2 df-nel 3037 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
3 eleq1 2813 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
43notbid 317 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
52, 4bitrid 282 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
61, 5mpbiri 257 . 2 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
7 funressndmafv2rn 46666 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
87con3i 154 . . . 4 (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
92, 8sylbi 216 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
10 ndfatafv2 46654 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
119, 10syl 17 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
126, 11impbii 208 1 ((𝐹''''𝐴) = 𝒫 ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1533  wcel 2098  wnel 3036  𝒫 cpw 4598   cuni 4903  ran crn 5673   defAt wdfat 46559  ''''cafv2 46651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-dfat 46562  df-afv2 46652
This theorem is referenced by: (None)
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