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Theorem fvbr0 6125
 Description: Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
fvbr0 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)

Proof of Theorem fvbr0
StepHypRef Expression
1 eqid 2610 . . . 4 (𝐹𝑋) = (𝐹𝑋)
2 tz6.12i 6124 . . . 4 ((𝐹𝑋) ≠ ∅ → ((𝐹𝑋) = (𝐹𝑋) → 𝑋𝐹(𝐹𝑋)))
31, 2mpi 20 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋𝐹(𝐹𝑋))
43necon1bi 2810 . 2 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
54orri 390 1 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475   ≠ wne 2780  ∅c0 3874   class class class wbr 4583  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  fvrn0  6126  eliman0  6133
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