Theorem osneisi 14875

Description: The non empty open sets are neighborhoods of the singletons.

Hypothesis

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

Assertion

Ref	Expression
osneisi	|- ((J e. Top /\ A =/= (/)) -> (A e. J -> A e. U_x e. X ((nei` J)` {x})))

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

Proof of Theorem osneisi

Step	Hyp	Ref	Expression
1		elisset 2299	. . . . . . 7 |- (J e. Top -> J e. _V)
2		ac5g 14388	. . . . . . 7 |- (J e. _V -> E.f(f Fn J /\ A.a e. J (a =/= (/) -> (f` a) e. a)))
3		neeq1 2024	. . . . . . . . . . . . . 14 |- (a = A -> (a =/= (/) <-> A =/= (/)))
4		fveq2 4681	. . . . . . . . . . . . . . 15 |- (a = A -> (f` a) = (f` A))
5		id 73	. . . . . . . . . . . . . . 15 |- (a = A -> a = A)
6	4, 5	eleq12d 1965	. . . . . . . . . . . . . 14 |- (a = A -> ((f` a) e. a <-> (f` A) e. A))
7	3, 6	imbi12d 688	. . . . . . . . . . . . 13 |- (a = A -> ((a =/= (/) -> (f` a) e. a) <-> (A =/= (/) -> (f` A) e. A)))
8	7	rcla4v 2376	. . . . . . . . . . . 12 |- (A e. J -> (A.a e. J (a =/= (/) -> (f` a) e. a) -> (A =/= (/) -> (f` A) e. A)))
9		pm2.27 76	. . . . . . . . . . . . . 14 |- (A =/= (/) -> ((A =/= (/) -> (f` A) e. A) -> (f` A) e. A))
10		elunii 3182	. . . . . . . . . . . . . . . 16 |- (((f` A) e. A /\ A e. J) -> (f` A) e. U.J)
11		sneq 3054	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x = (f` A) -> {x} = {(f` A)})
12	11	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x = (f` A) /\ y e. J) -> {x} = {(f` A)})
13	12	sseq1d 2644	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((x = (f` A) /\ y e. J) -> ({x} C_ y <-> {(f` A)} C_ y))
14	13	anbi1d 679	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((x = (f` A) /\ y e. J) -> (({x} C_ y /\ y C_ A) <-> ({(f` A)} C_ y /\ y C_ A)))
15	14	rexbidva 2120	. . . . . . . . . . . . . . . . . . . . . 22 |- (x = (f` A) -> (E.y e. J ({x} C_ y /\ y C_ A) <-> E.y e. J ({(f` A)} C_ y /\ y C_ A)))
16	15	anbi2d 678	. . . . . . . . . . . . . . . . . . . . 21 |- (x = (f` A) -> ((A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)) <-> (A C_ X /\ E.y e. J ({(f` A)} C_ y /\ y C_ A))))
17	16	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . 20 |- (((f` A) e. X /\ (A C_ X /\ E.y e. J ({(f` A)} C_ y /\ y C_ A))) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))
18	17	ex 402	. . . . . . . . . . . . . . . . . . 19 |- ((f` A) e. X -> ((A C_ X /\ E.y e. J ({(f` A)} C_ y /\ y C_ A)) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
19		elssuni 3206	. . . . . . . . . . . . . . . . . . . . . 22 |- (A e. J -> A C_ U.J)
20		osneisi.1	. . . . . . . . . . . . . . . . . . . . . 22 |- X = U.J
21	19, 20	syl6ssr 2664	. . . . . . . . . . . . . . . . . . . . 21 |- (A e. J -> A C_ X)
22	21	ad2antlr 441	. . . . . . . . . . . . . . . . . . . 20 |- ((((f` A) e. A /\ A e. J) /\ J e. Top) -> A C_ X)
23		snssi 3129	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((f` A) e. A -> {(f` A)} C_ A)
24		ssid 2634	. . . . . . . . . . . . . . . . . . . . . . 23 |- A C_ A
25	23, 24	jctir 317	. . . . . . . . . . . . . . . . . . . . . 22 |- ((f` A) e. A -> ({(f` A)} C_ A /\ A C_ A))
26		sseq2 2639	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = A -> ({(f` A)} C_ y <-> {(f` A)} C_ A))
27		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (y = A -> (y C_ A <-> A C_ A))
28	26, 27	anbi12d 690	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (y = A -> (({(f` A)} C_ y /\ y C_ A) <-> ({(f` A)} C_ A /\ A C_ A)))
29	28	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((A e. J /\ ({(f` A)} C_ A /\ A C_ A)) -> E.y e. J ({(f` A)} C_ y /\ y C_ A))
30	29	ex 402	. . . . . . . . . . . . . . . . . . . . . 22 |- (A e. J -> (({(f` A)} C_ A /\ A C_ A) -> E.y e. J ({(f` A)} C_ y /\ y C_ A)))
31	25, 30	mpan9 521	. . . . . . . . . . . . . . . . . . . . 21 |- (((f` A) e. A /\ A e. J) -> E.y e. J ({(f` A)} C_ y /\ y C_ A))
32	31	adantr 425	. . . . . . . . . . . . . . . . . . . 20 |- ((((f` A) e. A /\ A e. J) /\ J e. Top) -> E.y e. J ({(f` A)} C_ y /\ y C_ A))
33	22, 32	jca 310	. . . . . . . . . . . . . . . . . . 19 |- ((((f` A) e. A /\ A e. J) /\ J e. Top) -> (A C_ X /\ E.y e. J ({(f` A)} C_ y /\ y C_ A)))
34	18, 33	syl5com 63	. . . . . . . . . . . . . . . . . 18 |- ((((f` A) e. A /\ A e. J) /\ J e. Top) -> ((f` A) e. X -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
35	20	eqcomi 1888	. . . . . . . . . . . . . . . . . . 19 |- U.J = X
36	35	eleq2i 1961	. . . . . . . . . . . . . . . . . 18 |- ((f` A) e. U.J <-> (f` A) e. X)
37	34, 36	syl5ib 223	. . . . . . . . . . . . . . . . 17 |- ((((f` A) e. A /\ A e. J) /\ J e. Top) -> ((f` A) e. U.J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
38	37	ex 402	. . . . . . . . . . . . . . . 16 |- (((f` A) e. A /\ A e. J) -> (J e. Top -> ((f` A) e. U.J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))))
39	10, 38	mpid 58	. . . . . . . . . . . . . . 15 |- (((f` A) e. A /\ A e. J) -> (J e. Top -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
40	39	ex 402	. . . . . . . . . . . . . 14 |- ((f` A) e. A -> (A e. J -> (J e. Top -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))))
41	9, 40	syl6 25	. . . . . . . . . . . . 13 |- (A =/= (/) -> ((A =/= (/) -> (f` A) e. A) -> (A e. J -> (J e. Top -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
42	41	com4l 43	. . . . . . . . . . . 12 |- ((A =/= (/) -> (f` A) e. A) -> (A e. J -> (J e. Top -> (A =/= (/) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
43	8, 42	syl6 25	. . . . . . . . . . 11 |- (A e. J -> (A.a e. J (a =/= (/) -> (f` a) e. a) -> (A e. J -> (J e. Top -> (A =/= (/) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))))))
44	43	pm2.43a 80	. . . . . . . . . 10 |- (A e. J -> (A.a e. J (a =/= (/) -> (f` a) e. a) -> (J e. Top -> (A =/= (/) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
45	44	com4l 43	. . . . . . . . 9 |- (A.a e. J (a =/= (/) -> (f` a) e. a) -> (J e. Top -> (A =/= (/) -> (A e. J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
46	45	adantl 424	. . . . . . . 8 |- ((f Fn J /\ A.a e. J (a =/= (/) -> (f` a) e. a)) -> (J e. Top -> (A =/= (/) -> (A e. J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
47	46	19.23aiv 1674	. . . . . . 7 |- (E.f(f Fn J /\ A.a e. J (a =/= (/) -> (f` a) e. a)) -> (J e. Top -> (A =/= (/) -> (A e. J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
48	1, 2, 47	3syl 24	. . . . . 6 |- (J e. Top -> (J e. Top -> (A =/= (/) -> (A e. J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))))
49	48	pm2.43i 78	. . . . 5 |- (J e. Top -> (A =/= (/) -> (A e. J -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))))
50	49	imp31 389	. . . 4 |- (((J e. Top /\ A =/= (/)) /\ A e. J) -> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A)))
51	20	isnei 8994	. . . . . 6 |- ((J e. Top /\ {x} C_ X) -> (A e. ((nei` J)` {x}) <-> (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
52		simpll 448	. . . . . 6 |- (((J e. Top /\ A =/= (/)) /\ A e. J) -> J e. Top)
53		snssi 3129	. . . . . 6 |- (x e. X -> {x} C_ X)
54	51, 52, 53	syl2an 503	. . . . 5 |- ((((J e. Top /\ A =/= (/)) /\ A e. J) /\ x e. X) -> (A e. ((nei` J)` {x}) <-> (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
55	54	rexbidva 2120	. . . 4 |- (((J e. Top /\ A =/= (/)) /\ A e. J) -> (E.x e. X A e. ((nei` J)` {x}) <-> E.x e. X (A C_ X /\ E.y e. J ({x} C_ y /\ y C_ A))))
56	50, 55	mpbird 213	. . 3 |- (((J e. Top /\ A =/= (/)) /\ A e. J) -> E.x e. X A e. ((nei` J)` {x}))
57		eliun 3259	. . 3 |- (A e. U_x e. X ((nei` J)` {x}) <-> E.x e. X A e. ((nei` J)` {x}))
58	56, 57	sylibr 217	. 2 |- (((J e. Top /\ A =/= (/)) /\ A e. J) -> A e. U_x e. X ((nei` J)` {x}))
59	58	ex 402	1 |- ((J e. Top /\ A =/= (/)) -> (A e. J -> A e. U_x e. X ((nei` J)` {x})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  U_ciun 3255   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  ax-ac 5906

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-reu 2111  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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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