Theorem fvclss 6500

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 2629	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
2		tz6.12i 6214	. . . . . . . . . 10 |- ( y =/= (/) -> ( ( F ` x ) = y -> x F y ) )
3	1, 2	syl5bi 232	. . . . . . . . 9 |- ( y =/= (/) -> ( y = ( F ` x ) -> x F y ) )
4	3	eximdv 1846	. . . . . . . 8 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> E. x x F y ) )
5		vex 3203	. . . . . . . . 9 |- y e. _V
6	5	elrn 5366	. . . . . . . 8 |- ( y e. ran F <-> E. x x F y )
7	4, 6	syl6ibr 242	. . . . . . 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 2812	. . . . 5 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y = (/) ) )
10		velsn 4193	. . . . 5 |- ( y e. { (/) } <-> y = (/) )
11	9, 10	syl6ibr 242	. . . 4 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y e. { (/) } ) )
12	11	orrd 393	. . 3 |- ( E. x y = ( F ` x ) -> ( y e. ran F \/ y e. { (/) } ) )
13	12	ss2abi 3674	. 2 |- { y | E. x y = ( F ` x ) } C_ { y | ( y e. ran F \/ y e. { (/) } ) }
14		df-un 3579	. 2 |- ( ran F u. { (/) } ) = { y | ( y e. ran F \/ y e. { (/) } ) }
15	13, 14	sseqtr4i 3638	1 |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )

Syntax hints:   -. wn 3   \/ wo 383   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   ran crn 5115   ` 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-sep 4781  ax-nul 4789  ax-pr 4906

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-mo 2475  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-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvclex  7138