Theorem fvclss 3931

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

Assertion

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

Distinct variable group:   x,y,F

Proof of Theorem fvclss

Step	Hyp	Ref	Expression
1		visset 1851	. . . . . . . . . . 11 |- x e. V
2	1	tz6.12i 3817	. . . . . . . . . 10 |- (y =/= (/) -> ((F` x) = y -> xFy))
3		eqcom 1514	. . . . . . . . . 10 |- (y = (F` x) <-> (F` x) = y)
4	2, 3	syl5ib 204	. . . . . . . . 9 |- (y =/= (/) -> (y = (F` x) -> xFy))
5	4	19.22dv 1323	. . . . . . . 8 |- (y =/= (/) -> (E.x y = (F` x) -> E.x xFy))
6		visset 1851	. . . . . . . . 9 |- y e. V
7	6	elrn 3410	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
8	5, 7	syl6ibr 211	. . . . . . 7 |- (y =/= (/) -> (E.x y = (F` x) -> y e. ran F))
9	8	com12 11	. . . . . 6 |- (E.x y = (F` x) -> (y =/= (/) -> y e. ran F))
10	9	necon1bd 1669	. . . . 5 |- (E.x y = (F` x) -> (-. y e. ran F -> y = (/)))
11		elsn 2466	. . . . 5 |- (y e. {(/)} <-> y = (/))
12	10, 11	syl6ibr 211	. . . 4 |- (E.x y = (F` x) -> (-. y e. ran F -> y e. {(/)}))
13	12	orrd 231	. . 3 |- (E.x y = (F` x) -> (y e. ran F \/ y e. {(/)}))
14	13	ss2abi 2164	. 2 |- {y | E.x y = (F` x)} (_ {y | (y e. ran F \/ y e. {(/)})}
15		df-un 2094	. 2 |- (ran F u. {(/)}) = {y | (y e. ran F \/ y e. {(/)})}
16	14, 15	sseqtr4i 2138	1 |- {y | E.x y = (F` x)} (_ (ran F u. {(/)})

Syntax hints:  -. wn 2   \/ wo 220   = wceq 988   e. wcel 990  E.wex 1012  {cab 1499   =/= wne 1622   u. cun 2089   (_ wss 2091  (/)c0 2324  {csn 2454   class class class wbr 2669  ran crn 3226  ` cfv 3237

This theorem is referenced by:  fvclex 3932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-xp 3239  df-cnv 3241  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fv 3253

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