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Theorem ndfatafv2nrn 47171
Description: The alternate function value at a class 𝐴 at which the function is not defined is undefined, i.e., not in the range of the function. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndfatafv2nrn 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndfatafv2nrn
StepHypRef Expression
1 ndfatafv2 47161 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwuninel 8299 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
3 df-nel 3045 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4 eleq1 2827 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
54notbid 318 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
63, 5bitrid 283 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
72, 6mpbiri 258 . 2 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
81, 7syl 17 1 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wnel 3044  𝒫 cpw 4605   cuni 4912  ran crn 5690   defAt wdfat 47066  ''''cafv2 47158
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-uni 4913  df-afv2 47159
This theorem is referenced by:  ndmafv2nrn  47172  nfunsnafv2  47175  dfatafv2rnb  47177  tz6.12-2-afv2  47187
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