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Theorem afv2ndefb 42125
 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 7665 . . 3 ¬ 𝒫 ran 𝐹 ∈ ran 𝐹
2 df-nel 3102 . . . 4 ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ (𝐹''''𝐴) ∈ ran 𝐹)
3 eleq1 2893 . . . . 5 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∈ ran 𝐹 ↔ 𝒫 ran 𝐹 ∈ ran 𝐹))
43notbid 310 . . . 4 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (¬ (𝐹''''𝐴) ∈ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
52, 4syl5bb 275 . . 3 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → ((𝐹''''𝐴) ∉ ran 𝐹 ↔ ¬ 𝒫 ran 𝐹 ∈ ran 𝐹))
61, 5mpbiri 250 . 2 ((𝐹''''𝐴) = 𝒫 ran 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
7 funressndmafv2rn 42124 . . . . 5 (𝐹 defAt 𝐴 → (𝐹''''𝐴) ∈ ran 𝐹)
87con3i 152 . . . 4 (¬ (𝐹''''𝐴) ∈ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
92, 8sylbi 209 . . 3 ((𝐹''''𝐴) ∉ ran 𝐹 → ¬ 𝐹 defAt 𝐴)
10 ndfatafv2 42112 . . 3 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
119, 10syl 17 . 2 ((𝐹''''𝐴) ∉ ran 𝐹 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
126, 11impbii 201 1 ((𝐹''''𝐴) = 𝒫 ran 𝐹 ↔ (𝐹''''𝐴) ∉ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1658   ∈ wcel 2166   ∉ wnel 3101  𝒫 cpw 4377  ∪ cuni 4657  ran crn 5342   defAt wdfat 42017  ''''cafv2 42109 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126  ax-un 7208 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-nel 3102  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-iota 6085  df-fun 6124  df-dfat 42020  df-afv2 42110 This theorem is referenced by: (None)
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