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Theorem fvbr0 6854
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 2736 . . . 4 (𝐹𝑋) = (𝐹𝑋)
2 tz6.12i 6853 . . . 4 ((𝐹𝑋) ≠ ∅ → ((𝐹𝑋) = (𝐹𝑋) → 𝑋𝐹(𝐹𝑋)))
31, 2mpi 20 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋𝐹(𝐹𝑋))
43necon1bi 2969 . 2 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
54orri 859 1 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1540  wne 2940  c0 4269   class class class wbr 5092  cfv 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-nul 5250
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487
This theorem is referenced by:  fvrn0  6855  eliman0  6865
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