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Theorem fvbr0 6690
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 2818 . . . 4 (𝐹𝑋) = (𝐹𝑋)
2 tz6.12i 6689 . . . 4 ((𝐹𝑋) ≠ ∅ → ((𝐹𝑋) = (𝐹𝑋) → 𝑋𝐹(𝐹𝑋)))
31, 2mpi 20 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋𝐹(𝐹𝑋))
43necon1bi 3041 . 2 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
54orri 856 1 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wo 841   = wceq 1528  wne 3013  c0 4288   class class class wbr 5057  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  fvrn0  6691  eliman0  6698
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