Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvfv0bi Structured version   Visualization version   GIF version

Theorem afvfv0bi 44316
Description: The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfv0bi ((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))

Proof of Theorem afvfv0bi
StepHypRef Expression
1 ioran 984 . . . 4 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2 df-ne 2941 . . . . . . 7 ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3 afvnufveq 44311 . . . . . . 7 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
42, 3sylbir 238 . . . . . 6 (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2741 . . . . . . . 8 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 321 . . . . . . 7 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpd 232 . . . . . 6 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
84, 7syl 17 . . . . 5 (¬ (𝐹'''𝐴) = V → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
98impcom 411 . . . 4 ((¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
101, 9sylbi 220 . . 3 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
1110con4i 114 . 2 ((𝐹𝐴) = ∅ → ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
12 afv0fv0 44313 . . 3 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
13 afvpcfv0 44310 . . 3 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
1412, 13jaoi 857 . 2 (((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → (𝐹𝐴) = ∅)
1511, 14impbii 212 1 ((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wne 2940  Vcvv 3408  c0 4237  cfv 6380  '''cafv 44281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-res 5563  df-iota 6338  df-fun 6382  df-fv 6388  df-aiota 44249  df-dfat 44283  df-afv 44284
This theorem is referenced by:  aovov0bi  44360
  Copyright terms: Public domain W3C validator