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Theorem fvbr0 5693
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 2388 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
2 tz6.12i 5692 . . . 4  |-  ( ( F `  X )  =/=  (/)  ->  ( ( F `  X )  =  ( F `  X )  ->  X F ( F `  X ) ) )
31, 2mpi 17 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X F
( F `  X
) )
43necon1bi 2594 . 2  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
54orri 366 1  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
Colors of variables: wff set class
Syntax hints:    \/ wo 358    = wceq 1649    =/= wne 2551   (/)c0 3572   class class class wbr 4154   ` cfv 5395
This theorem is referenced by:  fvrn0  5694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-nul 4280
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-iota 5359  df-fv 5403
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