Theorem fvclss 6147

Description: Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)

Assertion

Ref	Expression
fvclss	|- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )

Distinct variable group:   x, y, F

Proof of Theorem fvclss

Step	Hyp	Ref	Expression
1		eqcom 2458	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
2		tz6.12i 5885	. . . . . . . . . 10 |- ( y =/= (/) -> ( ( F ` x ) = y -> x F y ) )
3	1, 2	syl5bi 221	. . . . . . . . 9 |- ( y =/= (/) -> ( y = ( F ` x ) -> x F y ) )
4	3	eximdv 1764	. . . . . . . 8 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> E. x x F y ) )
5		vex 3048	. . . . . . . . 9 |- y e. _V
6	5	elrn 5075	. . . . . . . 8 |- ( y e. ran F <-> E. x x F y )
7	4, 6	syl6ibr 231	. . . . . . 7 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> y e. ran F ) )
8	7	com12 32	. . . . . 6 |- ( E. x y = ( F ` x ) -> ( y =/= (/) -> y e. ran F ) )
9	8	necon1bd 2642	. . . . 5 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y = (/) ) )
10		elsn 3982	. . . . 5 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	syl6ibr 231	. . . 4 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y e. { (/) } ) )
12	11	orrd 380	. . 3 |- ( E. x y = ( F ` x ) -> ( y e. ran F \/ y e. { (/) } ) )
13	12	ss2abi 3501	. 2 |- { y | E. x y = ( F ` x ) } C_ { y | ( y e. ran F \/ y e. { (/) } ) }
14		df-un 3409	. 2 |- ( ran F u. { (/) } ) = { y | ( y e. ran F \/ y e. { (/) } ) }
15	13, 14	sseqtr4i 3465	1 |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )

Syntax hints:   -. wn 3   \/ wo 370   = wceq 1444   E.wex 1663   e. wcel 1887   {cab 2437   =/= wne 2622   u. cun 3402   C_ wss 3404   (/)c0 3731   {csn 3968   class class class wbr 4402   ran crn 4835   ` cfv 5582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-cnv 4842  df-dm 4844  df-rn 4845  df-iota 5546  df-fv 5590

This theorem is referenced by:  fvclex  6765