Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvpcfv0 Structured version   Visualization version   GIF version

Theorem afvpcfv0 41988
Description: If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvpcfv0 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)

Proof of Theorem afvpcfv0
StepHypRef Expression
1 dfafv2 41974 . . 3 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
21eqeq1i 2802 . 2 ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
3 eqcom 2804 . . . 4 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹𝐴), V))
4 eqif 4315 . . . 4 (V = if(𝐹 defAt 𝐴, (𝐹𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
53, 4bitri 267 . . 3 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
6 fveqvfvv 41911 . . . . . 6 ((𝐹𝐴) = V → (𝐹𝐴) = ∅)
76eqcoms 2805 . . . . 5 (V = (𝐹𝐴) → (𝐹𝐴) = ∅)
87adantl 474 . . . 4 ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) → (𝐹𝐴) = ∅)
9 fvfundmfvn0 6448 . . . . . . 7 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 41961 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
119, 10sylibr 226 . . . . . 6 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
1211necon1bi 2997 . . . . 5 𝐹 defAt 𝐴 → (𝐹𝐴) = ∅)
1312adantr 473 . . . 4 ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹𝐴) = ∅)
148, 13jaoi 884 . . 3 (((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹𝐴) = ∅)
155, 14sylbi 209 . 2 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V → (𝐹𝐴) = ∅)
162, 15sylbi 209 1 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2969  Vcvv 3383  c0 4113  ifcif 4275  {csn 4366  dom cdm 5310  cres 5312  Fun wfun 6093  cfv 6099   defAt wdfat 41958  '''cafv 41959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-res 5322  df-iota 6062  df-fun 6101  df-fv 6107  df-aiota 41922  df-dfat 41961  df-afv 41962
This theorem is referenced by:  afvfv0bi  41994  aovpcov0  42032
  Copyright terms: Public domain W3C validator