Theorem fvclss 4831

Description: Upper bound for the class of values of a class.

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		visset 2295	. . . . . . . . . . 11 |- x e. _V
2	1	tz6.12i 4698	. . . . . . . . . 10 |- (y =/= (/) -> ((F` x) = y -> xFy))
3		eqcom 1886	. . . . . . . . . 10 |- (y = (F` x) <-> (F` x) = y)
4	2, 3	syl5ib 223	. . . . . . . . 9 |- (y =/= (/) -> (y = (F` x) -> xFy))
5	4	eximdv 1669	. . . . . . . 8 |- (y =/= (/) -> (E.x y = (F` x) -> E.x xFy))
6		visset 2295	. . . . . . . . 9 |- y e. _V
7	6	elrn 4197	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
8	5, 7	syl6ibr 230	. . . . . . 7 |- (y =/= (/) -> (E.x y = (F` x) -> y e. ran F))
9	8	com12 14	. . . . . 6 |- (E.x y = (F` x) -> (y =/= (/) -> y e. ran F))
10	9	necon1bd 2080	. . . . 5 |- (E.x y = (F` x) -> (-. y e. ran F -> y = (/)))
11		elsn 3058	. . . . 5 |- (y e. {(/)} <-> y = (/))
12	10, 11	syl6ibr 230	. . . 4 |- (E.x y = (F` x) -> (-. y e. ran F -> y e. {(/)}))
13	12	orrd 250	. . 3 |- (E.x y = (F` x) -> (y e. ran F \/ y e. {(/)}))
14	13	ss2abi 2679	. 2 |- {y | E.x y = (F` x)} C_ {y | (y e. ran F \/ y e. {(/)})}
15		df-un 2600	. 2 |- (ran F u. {(/)}) = {y | (y e. ran F \/ y e. {(/)})}
16	14, 15	sseqtr4i 2650	1 |- {y | E.x y = (F` x)} C_ (ran F u. {(/)})

Syntax hints:  -. wn 2   \/ wo 239   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044   class class class wbr 3338  ran crn 3987  ` cfv 3998

This theorem is referenced by:  fvclex 4832

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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