Theorem tailuni 15638

Description: The union of all the tails of a directed set.

Hypothesis

Ref	Expression
tailuni.1	|- X = dom D

Assertion

Ref	Expression
tailuni	|- (D e. Dir -> U.ran (tail` D) = X)

Proof of Theorem tailuni

Step	Hyp	Ref	Expression
1		tailuni.1	. . . . . 6 |- X = dom D
2	1	tailmap 15636	. . . . 5 |- (D e. Dir -> (tail` D):X-->~PX)
3		frn 4569	. . . . . . 7 |- ((tail` D):X-->~PX -> ran (tail` D) C_ ~PX)
4	3	sseld 2619	. . . . . 6 |- ((tail` D):X-->~PX -> (x e. ran (tail` D) -> x e. ~PX))
5		visset 2295	. . . . . . 7 |- x e. _V
6	5	elpw 3037	. . . . . 6 |- (x e. ~PX <-> x C_ X)
7	4, 6	syl6ib 229	. . . . 5 |- ((tail` D):X-->~PX -> (x e. ran (tail` D) -> x C_ X))
8	2, 7	syl 12	. . . 4 |- (D e. Dir -> (x e. ran (tail` D) -> x C_ X))
9	8	r19.21aiv 2175	. . 3 |- (D e. Dir -> A.x e. ran (tail` D)x C_ X)
10		unissb 3208	. . 3 |- (U.ran (tail` D) C_ X <-> A.x e. ran (tail` D)x C_ X)
11	9, 10	sylibr 217	. 2 |- (D e. Dir -> U.ran (tail` D) C_ X)
12	1	tailini 15637	. . . . . 6 |- ((D e. Dir /\ x e. X) -> x e. ((tail` D)` x))
13		fnfvelrn 4786	. . . . . . 7 |- (((tail` D) Fn X /\ x e. X) -> ((tail` D)` x) e. ran (tail` D))
14		ffn 4562	. . . . . . . 8 |- ((tail` D):X-->~PX -> (tail` D) Fn X)
15	2, 14	syl 12	. . . . . . 7 |- (D e. Dir -> (tail` D) Fn X)
16	13, 15	sylan 497	. . . . . 6 |- ((D e. Dir /\ x e. X) -> ((tail` D)` x) e. ran (tail` D))
17		fvex 4689	. . . . . . 7 |- ((tail` D)` x) e. _V
18		eleq2 1958	. . . . . . . 8 |- (t = ((tail` D)` x) -> (x e. t <-> x e. ((tail` D)` x)))
19		eleq1 1957	. . . . . . . 8 |- (t = ((tail` D)` x) -> (t e. ran (tail` D) <-> ((tail` D)` x) e. ran (tail` D)))
20	18, 19	anbi12d 690	. . . . . . 7 |- (t = ((tail` D)` x) -> ((x e. t /\ t e. ran (tail` D)) <-> (x e. ((tail` D)` x) /\ ((tail` D)` x) e. ran (tail` D))))
21	17, 20	cla4ev 2371	. . . . . 6 |- ((x e. ((tail` D)` x) /\ ((tail` D)` x) e. ran (tail` D)) -> E.t(x e. t /\ t e. ran (tail` D)))
22	12, 16, 21	syl11anc 524	. . . . 5 |- ((D e. Dir /\ x e. X) -> E.t(x e. t /\ t e. ran (tail` D)))
23		eluni 3180	. . . . 5 |- (x e. U.ran (tail` D) <-> E.t(x e. t /\ t e. ran (tail` D)))
24	22, 23	sylibr 217	. . . 4 |- ((D e. Dir /\ x e. X) -> x e. U.ran (tail` D))
25	24	ex 402	. . 3 |- (D e. Dir -> (x e. X -> x e. U.ran (tail` D)))
26	25	ssrdv 2622	. 2 |- (D e. Dir -> X C_ U.ran (tail` D))
27	11, 26	eqssd 2633	1 |- (D e. Dir -> U.ran (tail` D) = X)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  dom cdm 3986  ran crn 3987   Fn wfn 3993  -->wf 3994  ` cfv 3998  Dircdir 10348  tailctail 10349

This theorem is referenced by:  filnet 15645

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-dir 10350  df-tail 10351

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