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Theorem fvbr0 5744
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 2435 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
2 tz6.12i 5743 . . . 4  |-  ( ( F `  X )  =/=  (/)  ->  ( ( F `  X )  =  ( F `  X )  ->  X F ( F `  X ) ) )
31, 2mpi 17 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X F
( F `  X
) )
43necon1bi 2641 . 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 1652    =/= wne 2598   (/)c0 3620   class class class wbr 4204   ` cfv 5446
This theorem is referenced by:  fvrn0  5745
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-nul 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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