Theorem fvclss 2907

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(s):   x,y,F

Proof of Theorem fvclss

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . . 11 |- x e. V
2	1	tz6.12i 2847	. . . . . . . . . 10 |- (-. y = (/) -> ((F` x) = y -> xFy))
3		cleqcom 1103	. . . . . . . . . 10 |- (y = (F` x) <-> (F` x) = y)
4	2, 3	syl5ib 181	. . . . . . . . 9 |- (-. y = (/) -> (y = (F` x) -> xFy))
5	4	19.22dv 947	. . . . . . . 8 |- (-. y = (/) -> (E.x y = (F` x) -> E.x xFy))
6		visset 1350	. . . . . . . . 9 |- y e. V
7	6	elrn2 2563	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
8	5, 7	syl6ibr 186	. . . . . . 7 |- (-. y = (/) -> (E.x y = (F` x) -> y e. ran F))
9	8	com12 13	. . . . . 6 |- (E.x y = (F` x) -> (-. y = (/) -> y e. ran F))
10	9	con1d 85	. . . . 5 |- (E.x y = (F` x) -> (-. y e. ran F -> y = (/)))
11		elsn 1820	. . . . 5 |- (y e. {(/)} <-> y = (/))
12	10, 11	syl6ibr 186	. . . 4 |- (E.x y = (F` x) -> (-. y e. ran F -> y e. {(/)}))
13	12	orrd 203	. . 3 |- (E.x y = (F` x) -> (y e. ran F \/ y e. {(/)}))
14	13	ss2abi 1552	. 2 |- {y | E.x y = (F` x)} (_ {y | (y e. ran F \/ y e. {(/)})}
15		df-un 1490	. 2 |- (ran F u. {(/)}) = {y | (y e. ran F \/ y e. {(/)})}
16	14, 15	sseqtr4 1533	1 |- {y | E.x y = (F` x)} (_ (ran F u. {(/)})

Syntax hints:  -. wn 1   \/ wo 195  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092   u. cun 1485   (_ wss 1487  (/)c0 1707  {csn 1808   class class class wbr 2054  ran crn 2411  ` cfv 2422

This theorem is referenced by:  fvclex 2908

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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