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Theorem afv2ndefb 43424
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 7940 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
2 df-nel 3124 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
3 eleq1 2900 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
43notbid 320 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
52, 4syl5bb 285 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
61, 5mpbiri 260 . 2 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
7 funressndmafv2rn 43423 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
87con3i 157 . . . 4 (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
92, 8sylbi 219 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
10 ndfatafv2 43411 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
119, 10syl 17 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
126, 11impbii 211 1 ((𝐹''''𝐴) = 𝒫 ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 208   = wceq 1533  wcel 2110  wnel 3123  𝒫 cpw 4538   cuni 4837  ran crn 5555   defAt wdfat 43316  ''''cafv2 43408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-iota 6313  df-fun 6356  df-dfat 43319  df-afv2 43409
This theorem is referenced by: (None)
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