Theorem sallnei 14873

Description: Two ways to state the set of all the neighborhoods.

Hypothesis

Ref	Expression
sallnei.1	|- X = U.J

Assertion

Ref	Expression
sallnei	|- (J e. Top -> U.ran (nei` J) = {v | (v C_ X /\ E.g e. J g C_ v)})

Distinct variable groups:   g,J,v   g,X,v

Proof of Theorem sallnei

Step	Hyp	Ref	Expression
1		sallnei.1	. . . 4 |- X = U.J
2	1	neif 8991	. . 3 |- (J e. Top -> (nei` J) Fn ~PX)
3		fniunfv 4841	. . . 4 |- ((nei` J) Fn ~PX -> U_s e. ~P X((nei` J)` s) = U.ran (nei` J))
4	3	eqcomd 1889	. . 3 |- ((nei` J) Fn ~PX -> U.ran (nei` J) = U_s e. ~P X((nei` J)` s))
5	2, 4	syl 12	. 2 |- (J e. Top -> U.ran (nei` J) = U_s e. ~P X((nei` J)` s))
6	1	neival 8993	. . . . . 6 |- ((J e. Top /\ s C_ X) -> ((nei` J)` s) = {v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))})
7	6	ex 402	. . . . 5 |- (J e. Top -> (s C_ X -> ((nei` J)` s) = {v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))}))
8		visset 2295	. . . . . 6 |- s e. _V
9	8	elpw 3037	. . . . 5 |- (s e. ~PX <-> s C_ X)
10	7, 9	syl5ib 223	. . . 4 |- (J e. Top -> (s e. ~PX -> ((nei` J)` s) = {v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))}))
11	10	r19.21aiv 2175	. . 3 |- (J e. Top -> A.s e. ~P X((nei` J)` s) = {v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))})
12		iuneq2 3273	. . 3 |- (A.s e. ~P X((nei` J)` s) = {v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))} -> U_s e. ~P X((nei` J)` s) = U_s e. ~P X{v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))})
13	11, 12	syl 12	. 2 |- (J e. Top -> U_s e. ~P X((nei` J)` s) = U_s e. ~P X{v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))})
14		simpr 350	. . . . . . . . . 10 |- ((s C_ g /\ g C_ v) -> g C_ v)
15	14	reximi 2198	. . . . . . . . 9 |- (E.g e. J (s C_ g /\ g C_ v) -> E.g e. J g C_ v)
16	15	anim2i 362	. . . . . . . 8 |- ((v C_ X /\ E.g e. J (s C_ g /\ g C_ v)) -> (v C_ X /\ E.g e. J g C_ v))
17	16	adantl 424	. . . . . . 7 |- ((s e. ~PX /\ (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))) -> (v C_ X /\ E.g e. J g C_ v))
18	17	r19.23aiva 2212	. . . . . 6 |- (E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)) -> (v C_ X /\ E.g e. J g C_ v))
19		0elpw 3473	. . . . . . . 8 |- (/) e. ~PX
20		sseq1 2637	. . . . . . . . . . . 12 |- (s = (/) -> (s C_ g <-> (/) C_ g))
21	20	anbi1d 679	. . . . . . . . . . 11 |- (s = (/) -> ((s C_ g /\ g C_ v) <-> ((/) C_ g /\ g C_ v)))
22	21	rexbidv 2124	. . . . . . . . . 10 |- (s = (/) -> (E.g e. J (s C_ g /\ g C_ v) <-> E.g e. J ((/) C_ g /\ g C_ v)))
23	22	anbi2d 678	. . . . . . . . 9 |- (s = (/) -> ((v C_ X /\ E.g e. J (s C_ g /\ g C_ v)) <-> (v C_ X /\ E.g e. J ((/) C_ g /\ g C_ v))))
24	23	rcla4ev 2381	. . . . . . . 8 |- (((/) e. ~PX /\ (v C_ X /\ E.g e. J ((/) C_ g /\ g C_ v))) -> E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)))
25	19, 24	mpan 759	. . . . . . 7 |- ((v C_ X /\ E.g e. J ((/) C_ g /\ g C_ v)) -> E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)))
26		0ss 2900	. . . . . . . . 9 |- (/) C_ g
27	26	biantrur 794	. . . . . . . 8 |- (g C_ v <-> ((/) C_ g /\ g C_ v))
28	27	rexbii 2128	. . . . . . 7 |- (E.g e. J g C_ v <-> E.g e. J ((/) C_ g /\ g C_ v))
29	25, 28	sylan2b 501	. . . . . 6 |- ((v C_ X /\ E.g e. J g C_ v) -> E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)))
30	18, 29	impbii 174	. . . . 5 |- (E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)) <-> (v C_ X /\ E.g e. J g C_ v))
31	30	a1i 8	. . . 4 |- (J e. Top -> (E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v)) <-> (v C_ X /\ E.g e. J g C_ v)))
32	31	abbidv 2008	. . 3 |- (J e. Top -> {v | E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v))} = {v | (v C_ X /\ E.g e. J g C_ v)})
33		iunab 3300	. . 3 |- U_s e. ~P X{v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))} = {v | E.s e. ~P X(v C_ X /\ E.g e. J (s C_ g /\ g C_ v))}
34	32, 33	syl5eq 1940	. 2 |- (J e. Top -> U_s e. ~P X{v | (v C_ X /\ E.g e. J (s C_ g /\ g C_ v))} = {v | (v C_ X /\ E.g e. J g C_ v)})
35	5, 13, 34	3eqtrd 1929	1 |- (J e. Top -> U.ran (nei` J) = {v | (v C_ X /\ E.g e. J g C_ v)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177  U_ciun 3255  ran crn 3987   Fn wfn 3993  ` cfv 3998  Topctop 8857  neicnei 8988

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-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-rab 2112  df-v 2294  df-sbc 2454  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-iun 3257  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-nei 8989

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