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Theorem fvbr0 5549
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 2283 . . . 4  |-  ( F `
 X )  =  ( F `  X
)
2 tz6.12i 5548 . . . 4  |-  ( ( F `  X )  =/=  (/)  ->  ( ( F `  X )  =  ( F `  X )  ->  X F ( F `  X ) ) )
31, 2mpi 16 . . 3  |-  ( ( F `  X )  =/=  (/)  ->  X F
( F `  X
) )
43necon1bi 2489 . 2  |-  ( -.  X F ( F `
 X )  -> 
( F `  X
)  =  (/) )
54orri 365 1  |-  ( X F ( F `  X )  \/  ( F `  X )  =  (/) )
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    =/= wne 2446   (/)c0 3455   class class class wbr 4023   ` cfv 5255
This theorem is referenced by:  fvrn0  5550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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