Theorem fvclss 6142

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 2476	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
2		tz6.12i 5886	. . . . . . . . . 10 |- ( y =/= (/) -> ( ( F ` x ) = y -> x F y ) )
3	1, 2	syl5bi 217	. . . . . . . . 9 |- ( y =/= (/) -> ( y = ( F ` x ) -> x F y ) )
4	3	eximdv 1686	. . . . . . . 8 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> E. x x F y ) )
5		vex 3116	. . . . . . . . 9 |- y e. _V
6	5	elrn 5243	. . . . . . . 8 |- ( y e. ran F <-> E. x x F y )
7	4, 6	syl6ibr 227	. . . . . . 7 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> y e. ran F ) )
8	7	com12 31	. . . . . 6 |- ( E. x y = ( F ` x ) -> ( y =/= (/) -> y e. ran F ) )
9	8	necon1bd 2685	. . . . 5 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y = (/) ) )
10		elsn 4041	. . . . 5 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	syl6ibr 227	. . . 4 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y e. { (/) } ) )
12	11	orrd 378	. . 3 |- ( E. x y = ( F ` x ) -> ( y e. ran F \/ y e. { (/) } ) )
13	12	ss2abi 3572	. 2 |- { y | E. x y = ( F ` x ) } C_ { y | ( y e. ran F \/ y e. { (/) } ) }
14		df-un 3481	. 2 |- ( ran F u. { (/) } ) = { y | ( y e. ran F \/ y e. { (/) } ) }
15	13, 14	sseqtr4i 3537	1 |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )

Syntax hints:   -. wn 3   \/ wo 368   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   u. cun 3474   C_ wss 3476   (/)c0 3785   {csn 4027   class class class wbr 4447   ran crn 5000   ` cfv 5588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-cnv 5007  df-dm 5009  df-rn 5010  df-iota 5551  df-fv 5596

This theorem is referenced by:  fvclex  6756