Theorem elunirnALT 3875

Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 3874, it doesn't appeal to ndmfv 3751 (via funiunfv 3872).

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 3548	. . . . . . . 8 |- (Fun F <-> F Fn dom F)
2		fvelrnb 3766	. . . . . . . 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 618	. . . . . 6 |- (Fun F -> ((A e. y /\ y e. ran F) <-> (A e. y /\ E.x e. dom F(F` x) = y)))
5		r19.42v 1767	. . . . . 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 540	. . . . 5 |- (Fun F -> ((A e. y /\ y e. ran F) <-> E.x e. dom F(A e. y /\ (F` x) = y)))
7		eleq2 1538	. . . . . . 7 |- ((F` x) = y -> (A e. (F` x) <-> A e. y))
8	7	biimparc 421	. . . . . 6 |- ((A e. y /\ (F` x) = y) -> A e. (F` x))
9	8	r19.22si 1737	. . . . 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 1216	. . 3 |- (Fun F -> (E.y(A e. y /\ y e. ran F) -> E.x e. dom F A e. (F` x)))
12		fvelrn 3818	. . . . . . 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 3738	. . . . . 6 |- (F` x) e. V
16		eleq2 1538	. . . . . . 7 |- (y = (F` x) -> (A e. y <-> A e. (F` x)))
17		eleq1 1537	. . . . . . 7 |- (y = (F` x) -> (y e. ran F <-> (F` x) e. ran F))
18	16, 17	anbi12d 630	. . . . . 6 |- (y = (F` x) -> ((A e. y /\ y e. ran F) <-> (A e. (F` x) /\ (F` x) e. ran F)))
19	15, 18	cla4ev 1872	. . . . 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 1750	. . 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 518	. 2 |- (Fun F -> (E.y(A e. y /\ y e. ran F) <-> E.x e. dom F A e. (F` x)))
23		eluni 2510	. 2 |- (A e. U.ran F <-> E.y(A e. y /\ y e. ran F))
24	22, 23	syl5bb 534	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 958   e. wcel 960  E.wex 982  E.wrex 1649  U.cuni 2507  dom cdm 3176  ran crn 3177  Fun wfun 3182   Fn wfn 3183  ` cfv 3188

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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