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Theorem ndmafv 47040
Description: The value of a class outside its domain is the universe, compare with ndmfv 6936. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
ndmafv 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)

Proof of Theorem ndmafv
StepHypRef Expression
1 df-dfat 47019 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
21simplbi 497 . 2 (𝐹 defAt 𝐴𝐴 ∈ dom 𝐹)
3 afvnfundmuv 47039 . 2 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
42, 3nsyl5 159 1 𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1535  wcel 2104  Vcvv 3477  {csn 4630  dom cdm 5683  cres 5685  Fun wfun 6552   defAt wdfat 47016  '''cafv 47017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-int 4954  df-br 5150  df-opab 5212  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-res 5695  df-iota 6510  df-fun 6560  df-fv 6566  df-aiota 46985  df-dfat 47019  df-afv 47020
This theorem is referenced by:  afvvdm  47041  afvprc  47044  afvco2  47076  ndmaov  47083  aovprc  47088
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