Theorem fvclss 3846

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 1809	. . . . . . . . . . 11 |- x e. V
2	1	tz6.12i 3732	. . . . . . . . . 10 |- (y =/= (/) -> ((F` x) = y -> xFy))
3		eqcom 1474	. . . . . . . . . 10 |- (y = (F` x) <-> (F` x) = y)
4	2, 3	syl5ib 206	. . . . . . . . 9 |- (y =/= (/) -> (y = (F` x) -> xFy))
5	4	19.22dv 1288	. . . . . . . 8 |- (y =/= (/) -> (E.x y = (F` x) -> E.x xFy))
6		visset 1809	. . . . . . . . 9 |- y e. V
7	6	elrn 3344	. . . . . . . 8 |- (y e. ran F <-> E.x xFy)
8	5, 7	syl6ibr 213	. . . . . . 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 1629	. . . . 5 |- (E.x y = (F` x) -> (-. y e. ran F -> y = (/)))
11		elsn 2417	. . . . 5 |- (y e. {(/)} <-> y = (/))
12	10, 11	syl6ibr 213	. . . 4 |- (E.x y = (F` x) -> (-. y e. ran F -> y e. {(/)}))
13	12	orrd 233	. . 3 |- (E.x y = (F` x) -> (y e. ran F \/ y e. {(/)}))
14	13	ss2abi 2116	. 2 |- {y | E.x y = (F` x)} (_ {y | (y e. ran F \/ y e. {(/)})}
15		df-un 2046	. 2 |- (ran F u. {(/)}) = {y | (y e. ran F \/ y e. {(/)})}
16	14, 15	sseqtr4 2090	1 |- {y | E.x y = (F` x)} (_ (ran F u. {(/)})

Syntax hints:  -. wn 2   \/ wo 222   = wceq 954   e. wcel 956  E.wex 978  {cab 1461   =/= wne 1582   u. cun 2041   (_ wss 2043  (/)c0 2276  {csn 2405   class class class wbr 2614  ran crn 3166  ` cfv 3177

This theorem is referenced by:  fvclex 3847

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

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