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Theorem fvbr0 6688
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 2824 . . . 4 (𝐹𝑋) = (𝐹𝑋)
2 tz6.12i 6687 . . . 4 ((𝐹𝑋) ≠ ∅ → ((𝐹𝑋) = (𝐹𝑋) → 𝑋𝐹(𝐹𝑋)))
31, 2mpi 20 . . 3 ((𝐹𝑋) ≠ ∅ → 𝑋𝐹(𝐹𝑋))
43necon1bi 3042 . 2 𝑋𝐹(𝐹𝑋) → (𝐹𝑋) = ∅)
54orri 859 1 (𝑋𝐹(𝐹𝑋) ∨ (𝐹𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1538  wne 3014  c0 4276   class class class wbr 5052  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351
This theorem is referenced by:  fvrn0  6689  eliman0  6696
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