Theorem istail 15634

Description: The tail of an element in a directed set.

Hypothesis

Ref	Expression
istail.1	|- X = dom D

Assertion

Ref	Expression
istail	|- ((D e. Dir /\ A e. X) -> ((tail` D)` A) = {x | ADx})

Distinct variable groups:   x,A   x,D   x,X

Proof of Theorem istail

Step	Hyp	Ref	Expression
1		istail.1	. . . . 5 |- X = dom D
2	1	tailf 15633	. . . 4 |- (D e. Dir -> (tail` D) = {<.y, z>. | (y e. X /\ z = {x | yDx})})
3	2	fveq1d 4683	. . 3 |- (D e. Dir -> ((tail` D)` A) = ({<.y, z>. | (y e. X /\ z = {x | yDx})}` A))
4	3	adantr 425	. 2 |- ((D e. Dir /\ A e. X) -> ((tail` D)` A) = ({<.y, z>. | (y e. X /\ z = {x | yDx})}` A))
5		visset 2295	. . . . . . . . . 10 |- x e. _V
6		brelrng 4190	. . . . . . . . . 10 |- ((A e. X /\ x e. _V /\ ADx) -> x e. ran D)
7	5, 6	mp3an2 1179	. . . . . . . . 9 |- ((A e. X /\ ADx) -> x e. ran D)
8	7	ex 402	. . . . . . . 8 |- (A e. X -> (ADx -> x e. ran D))
9	8	abssdv 2681	. . . . . . 7 |- (A e. X -> {x | ADx} C_ ran D)
10		ssun2 2768	. . . . . . . . 9 |- ran D C_ (dom D u. ran D)
11		dmrnssfld 4205	. . . . . . . . 9 |- (dom D u. ran D) C_ U.U.D
12	10, 11	sstri 2626	. . . . . . . 8 |- ran D C_ U.U.D
13	12	a1i 8	. . . . . . 7 |- (A e. X -> ran D C_ U.U.D)
14	9, 13	sstrd 2627	. . . . . 6 |- (A e. X -> {x | ADx} C_ U.U.D)
15	14	adantl 424	. . . . 5 |- ((D e. Dir /\ A e. X) -> {x | ADx} C_ U.U.D)
16		uniexg 3795	. . . . . . 7 |- (D e. Dir -> U.D e. _V)
17		uniexg 3795	. . . . . . 7 |- (U.D e. _V -> U.U.D e. _V)
18	16, 17	syl 12	. . . . . 6 |- (D e. Dir -> U.U.D e. _V)
19	18	adantr 425	. . . . 5 |- ((D e. Dir /\ A e. X) -> U.U.D e. _V)
20		ssexg 3457	. . . . 5 |- (({x | ADx} C_ U.U.D /\ U.U.D e. _V) -> {x | ADx} e. _V)
21	15, 19, 20	syl11anc 524	. . . 4 |- ((D e. Dir /\ A e. X) -> {x | ADx} e. _V)
22		simpr 350	. . . . . 6 |- ((y e. X /\ z = {x | yDx}) -> z = {x | yDx})
23	22	ssopab2i 3574	. . . . 5 |- {<.y, z>. | (y e. X /\ z = {x | yDx})} C_ {<.y, z>. | z = {x | yDx}}
24		funopabeq 4456	. . . . 5 |- Fun {<.y, z>. | z = {x | yDx}}
25		funss 4439	. . . . 5 |- ({<.y, z>. | (y e. X /\ z = {x | yDx})} C_ {<.y, z>. | z = {x | yDx}} -> (Fun {<.y, z>. | z = {x | yDx}} -> Fun {<.y, z>. | (y e. X /\ z = {x | yDx})}))
26	23, 24, 25	mp2 54	. . . 4 |- Fun {<.y, z>. | (y e. X /\ z = {x | yDx})}
27	21, 26	jctir 317	. . 3 |- ((D e. Dir /\ A e. X) -> ({x | ADx} e. _V /\ Fun {<.y, z>. | (y e. X /\ z = {x | yDx})}))
28		eqid 1884	. . . . . 6 |- {x | ADx} = {x | ADx}
29	28	jctr 315	. . . . 5 |- (A e. X -> (A e. X /\ {x | ADx} = {x | ADx}))
30	29	adantl 424	. . . 4 |- ((D e. Dir /\ A e. X) -> (A e. X /\ {x | ADx} = {x | ADx}))
31	20, 14, 18	syl2an 503	. . . . . 6 |- ((A e. X /\ D e. Dir) -> {x | ADx} e. _V)
32		eleq1 1957	. . . . . . . 8 |- (y = A -> (y e. X <-> A e. X))
33		breq1 3341	. . . . . . . . . 10 |- (y = A -> (yDx <-> ADx))
34	33	abbidv 2008	. . . . . . . . 9 |- (y = A -> {x | yDx} = {x | ADx})
35	34	eqeq2d 1895	. . . . . . . 8 |- (y = A -> (z = {x | yDx} <-> z = {x | ADx}))
36	32, 35	anbi12d 690	. . . . . . 7 |- (y = A -> ((y e. X /\ z = {x | yDx}) <-> (A e. X /\ z = {x | ADx})))
37		eqeq1 1890	. . . . . . . 8 |- (z = {x | ADx} -> (z = {x | ADx} <-> {x | ADx} = {x | ADx}))
38	37	anbi2d 678	. . . . . . 7 |- (z = {x | ADx} -> ((A e. X /\ z = {x | ADx}) <-> (A e. X /\ {x | ADx} = {x | ADx})))
39	36, 38	opelopabg 3567	. . . . . 6 |- ((A e. X /\ {x | ADx} e. _V) -> (<.A, {x | ADx}>. e. {<.y, z>. | (y e. X /\ z = {x | yDx})} <-> (A e. X /\ {x | ADx} = {x | ADx})))
40	31, 39	syldan 516	. . . . 5 |- ((A e. X /\ D e. Dir) -> (<.A, {x | ADx}>. e. {<.y, z>. | (y e. X /\ z = {x | yDx})} <-> (A e. X /\ {x | ADx} = {x | ADx})))
41	40	ancoms 484	. . . 4 |- ((D e. Dir /\ A e. X) -> (<.A, {x | ADx}>. e. {<.y, z>. | (y e. X /\ z = {x | yDx})} <-> (A e. X /\ {x | ADx} = {x | ADx})))
42	30, 41	mpbird 213	. . 3 |- ((D e. Dir /\ A e. X) -> <.A, {x | ADx}>. e. {<.y, z>. | (y e. X /\ z = {x | yDx})})
43		funopfvg 4711	. . 3 |- (({x | ADx} e. _V /\ Fun {<.y, z>. | (y e. X /\ z = {x | yDx})}) -> (<.A, {x | ADx}>. e. {<.y, z>. | (y e. X /\ z = {x | yDx})} -> ({<.y, z>. | (y e. X /\ z = {x | yDx})}` A) = {x | ADx}))
44	27, 42, 43	sylc 83	. 2 |- ((D e. Dir /\ A e. X) -> ({<.y, z>. | (y e. X /\ z = {x | yDx})}` A) = {x | ADx})
45	4, 44	eqtrd 1925	1 |- ((D e. Dir /\ A e. X) -> ((tail` D)` A) = {x | ADx})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  _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  ` cfv 3998  Dircdir 10348  tailctail 10349

This theorem is referenced by:  eltail 15635  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

Copyright terms: Public domain