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Theorem fvbr0 5877
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 2443 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
2 tz6.12i 5876 . . . 4  |-  ( ( F `  X )  =/=  (/)  ->  ( ( F `  X )  =  ( F `  X )  ->  X F ( F `  X ) ) )
31, 2mpi 17 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X F
( F `  X
) )
43necon1bi 2676 . 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 1383    =/= wne 2638   (/)c0 3770   class class class wbr 4437   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-nul 4566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-iota 5541  df-fv 5586
This theorem is referenced by:  fvrn0  5878  eliman0  5885
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