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Theorem ndfatafv2undef 43811
 Description: The alternate function value at a class 𝐴 is undefined if the function, whose range is a set, is not defined at 𝐴. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndfatafv2undef ((ran 𝐹𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))

Proof of Theorem ndfatafv2undef
StepHypRef Expression
1 ndfatafv2 43810 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) = 𝒫 ran 𝐹)
2 undefval 7928 . . 3 (ran 𝐹𝑉 → (Undef‘ran 𝐹) = 𝒫 ran 𝐹)
32eqcomd 2804 . 2 (ran 𝐹𝑉 → 𝒫 ran 𝐹 = (Undef‘ran 𝐹))
41, 3sylan9eqr 2855 1 ((ran 𝐹𝑉 ∧ ¬ 𝐹 defAt 𝐴) → (𝐹''''𝐴) = (Undef‘ran 𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  𝒫 cpw 4497  ∪ cuni 4801  ran crn 5521  ‘cfv 6325  Undefcund 7924   defAt wdfat 43715  ''''cafv2 43807 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-undef 7925  df-afv2 43808 This theorem is referenced by: (None)
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