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Theorem afv2ndefb 45530
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 8211 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
2 df-nel 3051 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
3 eleq1 2826 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
43notbid 318 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
52, 4bitrid 283 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
61, 5mpbiri 258 . 2 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
7 funressndmafv2rn 45529 . . . . 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 45517 . . 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 1542  wcel 2107  wnel 3050  𝒫 cpw 4565   cuni 4870  ran crn 5639   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-dfat 45425  df-afv2 45515
This theorem is referenced by: (None)
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