Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ndmafv2nrn Structured version   Visualization version   GIF version

Theorem ndmafv2nrn 43428
 Description: The value of a class outside its domain is not in the range, compare with ndmfv 6703. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
ndmafv2nrn 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem ndmafv2nrn
StepHypRef Expression
1 orc 863 . . 3 𝐴 ∈ dom 𝐹 → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2 ianor 978 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 43325 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3xchnxbir 335 . . 3 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
51, 4sylibr 236 . 2 𝐴 ∈ dom 𝐹 → ¬ 𝐹 defAt 𝐴)
6 ndfatafv2nrn 43427 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
75, 6syl 17 1 𝐴 ∈ dom 𝐹 → (𝐹''''𝐴) ∉ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   ∈ wcel 2113   ∉ wnel 3126  {csn 4570  dom cdm 5558  ran crn 5559   ↾ cres 5560  Fun wfun 6352   defAt wdfat 43322  ''''cafv2 43414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333  ax-un 7464 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-nel 3127  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-uni 4842  df-dfat 43325  df-afv2 43415 This theorem is referenced by:  afv2prc  43432  fafv2elrnb  43441
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