Theorem elunirnALT 3869

Description: Membership in the union of the range of a function, proved directly. Unlike elunirn 3868, it doesn't appeal to ndmfv 3745 (via funiunfv 3866).

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 3542	. . . . . . . 8 |- (Fun F <-> F Fn dom F)
2		fvelrnb 3760	. . . . . . . 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 616	. . . . . 6 |- (Fun F -> ((A e. y /\ y e. ran F) <-> (A e. y /\ E.x e. dom F(F` x) = y)))
5		r19.42v 1764	. . . . . 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 538	. . . . 5 |- (Fun F -> ((A e. y /\ y e. ran F) <-> E.x e. dom F(A e. y /\ (F` x) = y)))
7		eleq2 1535	. . . . . . 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 1734	. . . . 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 1214	. . 3 |- (Fun F -> (E.y(A e. y /\ y e. ran F) -> E.x e. dom F A e. (F` x)))
12		fvelrn 3812	. . . . . . 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 3732	. . . . . 6 |- (F` x) e. V
16		eleq2 1535	. . . . . . 7 |- (y = (F` x) -> (A e. y <-> A e. (F` x)))
17		eleq1 1534	. . . . . . 7 |- (y = (F` x) -> (y e. ran F <-> (F` x) e. ran F))
18	16, 17	anbi12d 628	. . . . . 6 |- (y = (F` x) -> ((A e. y /\ y e. ran F) <-> (A e. (F` x) /\ (F` x) e. ran F)))
19	15, 18	cla4ev 1869	. . . . 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 1747	. . 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 516	. 2 |- (Fun F -> (E.y(A e. y /\ y e. ran F) <-> E.x e. dom F A e. (F` x)))
23		eluni 2506	. 2 |- (A e. U.ran F <-> E.y(A e. y /\ y e. ran F))
24	22, 23	syl5bb 532	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 956   e. wcel 958  E.wex 980  E.wrex 1646  U.cuni 2503  dom cdm 3170  ran crn 3171  Fun wfun 3176   Fn wfn 3177  ` cfv 3182

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198

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