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Theorem afvfv0bi 47102
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 985 . . . 4 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2 df-ne 2939 . . . . . . 7 ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3 afvnufveq 47097 . . . . . . 7 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
42, 3sylbir 235 . . . . . 6 (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2739 . . . . . . . 8 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 318 . . . . . . 7 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ ↔ ¬ (𝐹𝐴) = ∅))
76biimpd 229 . . . . . 6 ((𝐹'''𝐴) = (𝐹𝐴) → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
84, 7syl 17 . . . . 5 (¬ (𝐹'''𝐴) = V → (¬ (𝐹'''𝐴) = ∅ → ¬ (𝐹𝐴) = ∅))
98impcom 407 . . . 4 ((¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
101, 9sylbi 217 . . 3 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → ¬ (𝐹𝐴) = ∅)
1110con4i 114 . 2 ((𝐹𝐴) = ∅ → ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
12 afv0fv0 47099 . . 3 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
13 afvpcfv0 47096 . . 3 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
1412, 13jaoi 857 . 2 (((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) → (𝐹𝐴) = ∅)
1511, 14impbii 209 1 ((𝐹𝐴) = ∅ ↔ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wne 2938  Vcvv 3478  c0 4339  cfv 6563  '''cafv 47067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070
This theorem is referenced by:  aovov0bi  47146
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