Theorem elunirnALT 3854

Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 3853, it doesn't appeal to ndmfv 3730 (via funiunfv 3851).

Assertion

Ref	Expression
elunirnALT	|- (Fun F -> (A e. U.ran F <-> E.x e. dom F A e. (F` x)))

Distinct variable groups:   x,A   x,F

Proof of Theorem elunirnALT

Step	Hyp	Ref	Expression
1		funfn 3528	. . . . . . . 8 |- (Fun F <-> F Fn dom F)
2		fvelrnb 3745	. . . . . . . 8 |- (F Fn dom F -> (y e. ran F <-> E.x e. dom F(F` x) = y))
3	1, 2	sylbi 199	. . . . . . 7 |- (Fun F -> (y e. ran F <-> E.x e. dom F(F` x) = y))
4	3	anbi2d 614	. . . . . 6 |- (Fun F -> ((A e. y /\ y e. ran F) <-> (A e. y /\ E.x e. dom F(F` x) = y)))
5		r19.42v 1756	. . . . . 6 |- (E.x e. dom F(A e. y /\ (F` x) = y) <-> (A e. y /\ E.x e. dom F(F` x) = y))
6	4, 5	syl6bbr 536	. . . . 5 |- (Fun F -> ((A e. y /\ y e. ran F) <-> E.x e. dom F(A e. y /\ (F` x) = y)))
7		eleq2 1527	. . . . . . 7 |- ((F` x) = y -> (A e. (F` x) <-> A e. y))
8	7	biimparc 419	. . . . . 6 |- ((A e. y /\ (F` x) = y) -> A e. (F` x))
9	8	r19.22si 1726	. . . . 5 |- (E.x e. dom F(A e. y /\ (F` x) = y) -> E.x e. dom F A e. (F` x))
10	6, 9	syl6bi 214	. . . 4 |- (Fun F -> ((A e. y /\ y e. ran F) -> E.x e. dom F A e. (F` x)))
11	10	19.23adv 1209	. . 3 |- (Fun F -> (E.y(A e. y /\ y e. ran F) -> E.x e. dom F A e. (F` x)))
12		fvelrn 3797	. . . . . . 7 |- ((Fun F /\ x e. dom F) -> (F` x) e. ran F)
13	12	a1d 12	. . . . . 6 |- ((Fun F /\ x e. dom F) -> (A e. (F` x) -> (F` x) e. ran F))
14	13	ancld 298	. . . . 5 |- ((Fun F /\ x e. dom F) -> (A e. (F` x) -> (A e. (F` x) /\ (F` x) e. ran F)))
15		fvex 3717	. . . . . 6 |- (F` x) e. V
16		eleq2 1527	. . . . . . 7 |- (y = (F` x) -> (A e. y <-> A e. (F` x)))
17		eleq1 1526	. . . . . . 7 |- (y = (F` x) -> (y e. ran F <-> (F` x) e. ran F))
18	16, 17	anbi12d 626	. . . . . 6 |- (y = (F` x) -> ((A e. y /\ y e. ran F) <-> (A e. (F` x) /\ (F` x) e. ran F)))
19	15, 18	cla4ev 1860	. . . . 5 |- ((A e. (F` x) /\ (F` x) e. ran F) -> E.y(A e. y /\ y e. ran F))
20	14, 19	syl6 22	. . . 4 |- ((Fun F /\ x e. dom F) -> (A e. (F` x) -> E.y(A e. y /\ y e. ran F)))
21	20	r19.23adva 1739	. . 3 |- (Fun F -> (E.x e. dom F A e. (F` x) -> E.y(A e. y /\ y e. ran F)))
22	11, 21	impbid 514	. 2 |- (Fun F -> (E.y(A e. y /\ y e. ran F) <-> E.x e. dom F A e. (F` x)))
23		eluni 2496	. 2 |- (A e. U.ran F <-> E.y(A e. y /\ y e. ran F))
24	22, 23	syl5bb 530	1 |- (Fun F -> (A e. U.ran F <-> E.x e. dom F A e. (F` x)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E.wrex 1638  U.cuni 2493  dom cdm 3160  ran crn 3161  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

Copyright terms: Public domain