Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nfunsnafv2 Structured version   Visualization version   GIF version

Theorem nfunsnafv2 43638
Description: If the restriction of a class to a singleton is not a function, its value at the singleton element is undefined, compare with nfunsn 6690. (Contributed by AV, 2-Sep-2022.)
Assertion
Ref Expression
nfunsnafv2 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)

Proof of Theorem nfunsnafv2
StepHypRef Expression
1 olc 865 . . 3 (¬ Fun (𝐹 ↾ {𝐴}) → (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
2 ianor 979 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
3 df-dfat 43532 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
42, 3xchnxbir 336 . . 3 𝐹 defAt 𝐴 ↔ (¬ 𝐴 ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {𝐴})))
51, 4sylibr 237 . 2 (¬ Fun (𝐹 ↾ {𝐴}) → ¬ 𝐹 defAt 𝐴)
6 ndfatafv2nrn 43634 . 2 𝐹 defAt 𝐴 → (𝐹''''𝐴) ∉ ran 𝐹)
75, 6syl 17 1 (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹''''𝐴) ∉ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wo 844  wcel 2115  wnel 3117  {csn 4548  dom cdm 5538  ran crn 5539  cres 5540  Fun wfun 6332   defAt wdfat 43529  ''''cafv2 43621
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313  ax-un 7446
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-nel 3118  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-uni 4822  df-dfat 43532  df-afv2 43622
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator