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Theorem ndfatafv2nrn 46660
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 46650 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 pwuninel 8274 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
3 df-nel 3037 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
4 eleq1 2813 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
54notbid 317 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
63, 5bitrid 282 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
72, 6mpbiri 257 . 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 1533  wcel 2098  wnel 3036  𝒫 cpw 4599   cuni 4904  ran crn 5674   defAt wdfat 46555  ''''cafv2 46647
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-ext 2696  ax-sep 5295  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nel 3037  df-rab 3420  df-v 3465  df-un 3946  df-in 3948  df-ss 3958  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4905  df-afv2 46648
This theorem is referenced by:  ndmafv2nrn  46661  nfunsnafv2  46664  dfatafv2rnb  46666  tz6.12-2-afv2  46676
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