Theorem fvclss 5964

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 2445	. . . . . . . . . 10 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
2		tz6.12i 5715	. . . . . . . . . 10 |- ( y =/= (/) -> ( ( F ` x ) = y -> x F y ) )
3	1, 2	syl5bi 217	. . . . . . . . 9 |- ( y =/= (/) -> ( y = ( F ` x ) -> x F y ) )
4	3	eximdv 1676	. . . . . . . 8 |- ( y =/= (/) -> ( E. x y = ( F ` x ) -> E. x x F y ) )
5		vex 2980	. . . . . . . . 9 |- y e. _V
6	5	elrn 5085	. . . . . . . 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 2684	. . . . 5 |- ( E. x y = ( F ` x ) -> ( -. y e. ran F -> y = (/) ) )
10		elsn 3896	. . . . 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 3429	. 2 |- { y | E. x y = ( F ` x ) } C_ { y | ( y e. ran F \/ y e. { (/) } ) }
14		df-un 3338	. 2 |- ( ran F u. { (/) } ) = { y | ( y e. ran F \/ y e. { (/) } ) }
15	13, 14	sseqtr4i 3394	1 |- { y | E. x y = ( F ` x ) } C_ ( ran F u. { (/) } )

Syntax hints:   -. wn 3   \/ wo 368   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   =/= wne 2611   u. cun 3331   C_ wss 3333   (/)c0 3642   {csn 3882   class class class wbr 4297   ran crn 4846   ` cfv 5423

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-cnv 4853  df-dm 4855  df-rn 4856  df-iota 5386  df-fv 5431

This theorem is referenced by:  fvclex  6554