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Theorem fvbr0 5887
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  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )

Proof of Theorem fvbr0
StepHypRef Expression
1 eqid 2467 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
2 tz6.12i 5886 . . . 4  |-  ( ( F `  X )  =/=  (/)  ->  ( ( F `  X )  =  ( F `  X )  ->  X F ( F `  X ) ) )
31, 2mpi 17 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X F
( F `  X
) )
43necon1bi 2700 . 2  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
54orri 376 1  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    = wceq 1379    =/= wne 2662   (/)c0 3785   class class class wbr 4447   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-iota 5551  df-fv 5596
This theorem is referenced by:  fvrn0  5888  eliman0  5895
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