Theorem tailf 15633

Description: The tail function for a directed set.

Hypothesis

Ref	Expression
tailf.1	|- X = dom D

Assertion

Ref	Expression
tailf	|- (D e. Dir -> (tail` D) = {<.x, y>. | (x e. X /\ y = {z | xDz})})

Distinct variable groups:   x,y,z,D   x,X,y

Proof of Theorem tailf

Step	Hyp	Ref	Expression
1		uniexg 3795	. . . . . . . . 9 |- (D e. Dir -> U.D e. _V)
2		uniexg 3795	. . . . . . . . 9 |- (U.D e. _V -> U.U.D e. _V)
3		ssun2 2768	. . . . . . . . . . . . . 14 |- ran D C_ (dom D u. ran D)
4		dmrnssfld 4205	. . . . . . . . . . . . . 14 |- (dom D u. ran D) C_ U.U.D
5	3, 4	sstri 2626	. . . . . . . . . . . . 13 |- ran D C_ U.U.D
6	5	a1i 8	. . . . . . . . . . . 12 |- (xDz -> ran D C_ U.U.D)
7		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
8		visset 2295	. . . . . . . . . . . . 13 |- z e. _V
9	7, 8	brelrn 4191	. . . . . . . . . . . 12 |- (xDz -> z e. ran D)
10	6, 9	sseldd 2620	. . . . . . . . . . 11 |- (xDz -> z e. U.U.D)
11	10	abssi 2682	. . . . . . . . . 10 |- {z | xDz} C_ U.U.D
12		ssexg 3457	. . . . . . . . . 10 |- (({z | xDz} C_ U.U.D /\ U.U.D e. _V) -> {z | xDz} e. _V)
13	11, 12	mpan 759	. . . . . . . . 9 |- (U.U.D e. _V -> {z | xDz} e. _V)
14	1, 2, 13	3syl 24	. . . . . . . 8 |- (D e. Dir -> {z | xDz} e. _V)
15	14	adantr 425	. . . . . . 7 |- ((D e. Dir /\ x e. X) -> {z | xDz} e. _V)
16	15	r19.21aiva 2176	. . . . . 6 |- (D e. Dir -> A.x e. X {z | xDz} e. _V)
17		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. X /\ y = {z | xDz})}
18	17	fnopab2g 4547	. . . . . 6 |- (A.x e. X {z | xDz} e. _V <-> {<.x, y>. | (x e. X /\ y = {z | xDz})} Fn X)
19	16, 18	sylib 215	. . . . 5 |- (D e. Dir -> {<.x, y>. | (x e. X /\ y = {z | xDz})} Fn X)
20		dmexg 4206	. . . . . 6 |- (D e. Dir -> dom D e. _V)
21		tailf.1	. . . . . 6 |- X = dom D
22	20, 21	syl5eqel 1975	. . . . 5 |- (D e. Dir -> X e. _V)
23		fnex 4535	. . . . 5 |- (({<.x, y>. | (x e. X /\ y = {z | xDz})} Fn X /\ X e. _V) -> {<.x, y>. | (x e. X /\ y = {z | xDz})} e. _V)
24	19, 22, 23	syl11anc 524	. . . 4 |- (D e. Dir -> {<.x, y>. | (x e. X /\ y = {z | xDz})} e. _V)
25		simpr 350	. . . . . 6 |- ((w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}) -> t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})
26	25	ssopab2i 3574	. . . . 5 |- {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})} C_ {<.w, t>. | t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}}
27		funopabeq 4456	. . . . 5 |- Fun {<.w, t>. | t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}}
28		funss 4439	. . . . 5 |- ({<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})} C_ {<.w, t>. | t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}} -> (Fun {<.w, t>. | t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}} -> Fun {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}))
29	26, 27, 28	mp2 54	. . . 4 |- Fun {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}
30	24, 29	jctir 317	. . 3 |- (D e. Dir -> ({<.x, y>. | (x e. X /\ y = {z | xDz})} e. _V /\ Fun {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}))
31		dirdm 10354	. . . . . . . . 9 |- (D e. Dir -> dom D = U.U.D)
32	31, 21	syl5eq 1940	. . . . . . . 8 |- (D e. Dir -> X = U.U.D)
33	32	eleq2d 1964	. . . . . . 7 |- (D e. Dir -> (x e. X <-> x e. U.U.D))
34	33	anbi1d 679	. . . . . 6 |- (D e. Dir -> ((x e. X /\ y = {z | xDz}) <-> (x e. U.U.D /\ y = {z | xDz})))
35	34	opabbidv 3401	. . . . 5 |- (D e. Dir -> {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})
36	35	ancli 320	. . . 4 |- (D e. Dir -> (D e. Dir /\ {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})}))
37		eleq1 1957	. . . . . . 7 |- (w = D -> (w e. Dir <-> D e. Dir))
38		unieq 3185	. . . . . . . . . . . 12 |- (w = D -> U.w = U.D)
39	38	unieqd 3188	. . . . . . . . . . 11 |- (w = D -> U.U.w = U.U.D)
40	39	eleq2d 1964	. . . . . . . . . 10 |- (w = D -> (x e. U.U.w <-> x e. U.U.D))
41		breq 3340	. . . . . . . . . . . 12 |- (w = D -> (xwz <-> xDz))
42	41	abbidv 2008	. . . . . . . . . . 11 |- (w = D -> {z | xwz} = {z | xDz})
43	42	eqeq2d 1895	. . . . . . . . . 10 |- (w = D -> (y = {z | xwz} <-> y = {z | xDz}))
44	40, 43	anbi12d 690	. . . . . . . . 9 |- (w = D -> ((x e. U.U.w /\ y = {z | xwz}) <-> (x e. U.U.D /\ y = {z | xDz})))
45	44	opabbidv 3401	. . . . . . . 8 |- (w = D -> {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})
46	45	eqeq2d 1895	. . . . . . 7 |- (w = D -> (t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})} <-> t = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})}))
47	37, 46	anbi12d 690	. . . . . 6 |- (w = D -> ((w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})}) <-> (D e. Dir /\ t = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})))
48		eqeq1 1890	. . . . . . 7 |- (t = {<.x, y>. | (x e. X /\ y = {z | xDz})} -> (t = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})} <-> {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})}))
49	48	anbi2d 678	. . . . . 6 |- (t = {<.x, y>. | (x e. X /\ y = {z | xDz})} -> ((D e. Dir /\ t = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})}) <-> (D e. Dir /\ {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})))
50	47, 49	opelopabg 3567	. . . . 5 |- ((D e. Dir /\ {<.x, y>. | (x e. X /\ y = {z | xDz})} e. _V) -> (<.D, {<.x, y>. | (x e. X /\ y = {z | xDz})}>. e. {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})} <-> (D e. Dir /\ {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})))
51	24, 50	mpdan 768	. . . 4 |- (D e. Dir -> (<.D, {<.x, y>. | (x e. X /\ y = {z | xDz})}>. e. {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})} <-> (D e. Dir /\ {<.x, y>. | (x e. X /\ y = {z | xDz})} = {<.x, y>. | (x e. U.U.D /\ y = {z | xDz})})))
52	36, 51	mpbird 213	. . 3 |- (D e. Dir -> <.D, {<.x, y>. | (x e. X /\ y = {z | xDz})}>. e. {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})})
53		funopfvg 4711	. . 3 |- (({<.x, y>. | (x e. X /\ y = {z | xDz})} e. _V /\ Fun {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}) -> (<.D, {<.x, y>. | (x e. X /\ y = {z | xDz})}>. e. {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})} -> ({<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}` D) = {<.x, y>. | (x e. X /\ y = {z | xDz})}))
54	30, 52, 53	sylc 83	. 2 |- (D e. Dir -> ({<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}` D) = {<.x, y>. | (x e. X /\ y = {z | xDz})})
55		df-tail 10351	. . 3 |- tail = {<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}
56	55	fveq1i 4682	. 2 |- (tail` D) = ({<.w, t>. | (w e. Dir /\ t = {<.x, y>. | (x e. U.U.w /\ y = {z | xwz})})}` D)
57	54, 56	syl5eq 1940	1 |- (D e. Dir -> (tail` D) = {<.x, y>. | (x e. X /\ y = {z | xDz})})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  _Vcvv 2292   u. cun 2591   C_ wss 2593  <.cop 3046  U.cuni 3177   class class class wbr 3338  {copab 3395  dom cdm 3986  ran crn 3987  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  Dircdir 10348  tailctail 10349

This theorem is referenced by:  istail 15634  tailmap 15636

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-fv 4014  df-dir 10350  df-tail 10351

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