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Theorem afvfv0bi 45846
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 982 . . . 4 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2 df-ne 2941 . . . . . . 7 ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3 afvnufveq 45841 . . . . . . 7 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
42, 3sylbir 234 . . . . . 6 (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2736 . . . . . . . 8 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 317 . . . . . . 7 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpd 228 . . . . . 6 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
84, 7syl 17 . . . . 5 (¬ (𝐹'''𝐴) = V → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
98impcom 408 . . . 4 ((¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
101, 9sylbi 216 . . 3 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
1110con4i 114 . 2 ((𝐹𝐴) = ∅ → ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
12 afv0fv0 45843 . . 3 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
13 afvpcfv0 45840 . . 3 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
1412, 13jaoi 855 . 2 (((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → (𝐹𝐴) = ∅)
1511, 14impbii 208 1 ((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wne 2940  Vcvv 3474  c0 4321  cfv 6540  '''cafv 45811
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6492  df-fun 6542  df-fv 6548  df-aiota 45779  df-dfat 45813  df-afv 45814
This theorem is referenced by:  aovov0bi  45890
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