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Theorem afvfv0bi 41900
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 1006 . . . 4 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2 df-ne 2938 . . . . . . 7 ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3 afvnufveq 41895 . . . . . . 7 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
42, 3sylbir 226 . . . . . 6 (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2769 . . . . . . . 8 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 309 . . . . . . 7 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpd 220 . . . . . 6 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
84, 7syl 17 . . . . 5 (¬ (𝐹'''𝐴) = V → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
98impcom 396 . . . 4 ((¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
101, 9sylbi 208 . . 3 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
1110con4i 114 . 2 ((𝐹𝐴) = ∅ → ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
12 afv0fv0 41897 . . 3 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
13 afvpcfv0 41894 . . 3 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
1412, 13jaoi 883 . 2 (((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → (𝐹𝐴) = ∅)
1511, 14impbii 200 1 ((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873   = wceq 1652  wne 2937  Vcvv 3350  c0 4079  cfv 6068  '''cafv 41865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-aiota 41828  df-dfat 41867  df-afv 41868
This theorem is referenced by:  aovov0bi  41944
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