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Theorem afvfv0bi 45860
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 983 . . . 4 (¬ ((𝐹'''𝐴) = ∅ ∨ (𝐹'''𝐴) = V) ↔ (¬ (𝐹'''𝐴) = ∅ ∧ ¬ (𝐹'''𝐴) = V))
2 df-ne 2942 . . . . . . 7 ((𝐹'''𝐴) ≠ V ↔ ¬ (𝐹'''𝐴) = V)
3 afvnufveq 45855 . . . . . . 7 ((𝐹'''𝐴) ≠ V → (𝐹'''𝐴) = (𝐹𝐴))
42, 3sylbir 234 . . . . . 6 (¬ (𝐹'''𝐴) = V → (𝐹'''𝐴) = (𝐹𝐴))
5 eqeq1 2737 . . . . . . . 8 ((𝐹'''𝐴) = (𝐹𝐴) → ((𝐹'''𝐴) = ∅ ↔ (𝐹𝐴) = ∅))
65notbid 318 . . . . . . 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 409 . . . 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 45857 . . 3 ((𝐹'''𝐴) = ∅ → (𝐹𝐴) = ∅)
13 afvpcfv0 45854 . . 3 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
1412, 13jaoi 856 . 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 397  wo 846   = wceq 1542  wne 2941  Vcvv 3475  c0 4323  cfv 6544  '''cafv 45825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-aiota 45793  df-dfat 45827  df-afv 45828
This theorem is referenced by:  aovov0bi  45904
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