Theorem fvbr0 6215

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

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( F ` X ) = ( F ` X )
2		tz6.12i 6214	. . . 4 |- ( ( F ` X ) =/= (/) -> ( ( F ` X ) = ( F ` X ) -> X F ( F ` X ) ) )
3	1, 2	mpi 20	. . 3 |- ( ( F ` X ) =/= (/) -> X F ( F ` X ) )
4	3	necon1bi 2822	. 2 |- ( -. X F ( F ` X ) -> ( F ` X ) = (/) )
5	4	orri 391	1 |- ( X F ( F ` X ) \/ ( F ` X ) = (/) )

Syntax hints:   \/ wo 383   = wceq 1483   =/= wne 2794   (/)c0 3915   class class class wbr 4653   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvrn0  6216  eliman0  6223