Theorem subbas 8914

Description: The collection of finite intersections of the elements of any set A is a basis for a topology (on U.A by subbas2 8915). The set A is called a subbasis for that topology. Theorem for subbases in [Munkres] p. 82. See abfii1 5651 through abfii5 5655 for other ways to express the collection of finite intersections.

Hypotheses

Ref	Expression
subbas.1	|- A e. _V
subbas.2	|- B = {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}

Assertion

Ref	Expression
subbas	|- B e. Bases

Distinct variable group:   x,y,A

Proof of Theorem subbas

Step	Hyp	Ref	Expression
1		eeanv 1707	. . . . . 6 |- (E.fE.g((f C_ A /\ f e. Fin /\ w = |^|f) /\ (g C_ A /\ g e. Fin /\ v = |^|g)) <-> (E.f(f C_ A /\ f e. Fin /\ w = |^|f) /\ E.g(g C_ A /\ g e. Fin /\ v = |^|g)))
2		an6 1177	. . . . . . . 8 |- (((f C_ A /\ f e. Fin /\ w = |^|f) /\ (g C_ A /\ g e. Fin /\ v = |^|g)) <-> ((f C_ A /\ g C_ A) /\ (f e. Fin /\ g e. Fin) /\ (w = |^|f /\ v = |^|g)))
3		eleq2 1958	. . . . . . . . . . . 12 |- (t = (w i^i v) -> (u e. t <-> u e. (w i^i v)))
4		sseq1 2637	. . . . . . . . . . . 12 |- (t = (w i^i v) -> (t C_ (w i^i v) <-> (w i^i v) C_ (w i^i v)))
5	3, 4	anbi12d 690	. . . . . . . . . . 11 |- (t = (w i^i v) -> ((u e. t /\ t C_ (w i^i v)) <-> (u e. (w i^i v) /\ (w i^i v) C_ (w i^i v))))
6	5	rcla4ev 2381	. . . . . . . . . 10 |- (((w i^i v) e. B /\ (u e. (w i^i v) /\ (w i^i v) C_ (w i^i v))) -> E.t e. B (u e. t /\ t C_ (w i^i v)))
7		visset 2295	. . . . . . . . . . . . . 14 |- f e. _V
8		visset 2295	. . . . . . . . . . . . . 14 |- g e. _V
9	7, 8	unex 3796	. . . . . . . . . . . . 13 |- (f u. g) e. _V
10		sseq1 2637	. . . . . . . . . . . . . 14 |- (y = (f u. g) -> (y C_ A <-> (f u. g) C_ A))
11		eleq1 1957	. . . . . . . . . . . . . 14 |- (y = (f u. g) -> (y e. Fin <-> (f u. g) e. Fin))
12		inteq 3217	. . . . . . . . . . . . . . 15 |- (y = (f u. g) -> |^|y = |^|(f u. g))
13	12	eqeq2d 1895	. . . . . . . . . . . . . 14 |- (y = (f u. g) -> ((w i^i v) = |^|y <-> (w i^i v) = |^|(f u. g)))
14	10, 11, 13	3anbi123d 1168	. . . . . . . . . . . . 13 |- (y = (f u. g) -> ((y C_ A /\ y e. Fin /\ (w i^i v) = |^|y) <-> ((f u. g) C_ A /\ (f u. g) e. Fin /\ (w i^i v) = |^|(f u. g))))
15	9, 14	cla4ev 2371	. . . . . . . . . . . 12 |- (((f u. g) C_ A /\ (f u. g) e. Fin /\ (w i^i v) = |^|(f u. g)) -> E.y(y C_ A /\ y e. Fin /\ (w i^i v) = |^|y))
16		unss 2780	. . . . . . . . . . . . 13 |- ((f C_ A /\ g C_ A) <-> (f u. g) C_ A)
17	16	biimpi 168	. . . . . . . . . . . 12 |- ((f C_ A /\ g C_ A) -> (f u. g) C_ A)
18		unfi 5644	. . . . . . . . . . . 12 |- ((f e. Fin /\ g e. Fin) -> (f u. g) e. Fin)
19		ineq12 2791	. . . . . . . . . . . . 13 |- ((w = |^|f /\ v = |^|g) -> (w i^i v) = (|^|f i^i |^|g))
20		intun 3249	. . . . . . . . . . . . 13 |- |^|(f u. g) = (|^|f i^i |^|g)
21	19, 20	syl6eqr 1946	. . . . . . . . . . . 12 |- ((w = |^|f /\ v = |^|g) -> (w i^i v) = |^|(f u. g))
22	15, 17, 18, 21	syl3an 1139	. . . . . . . . . . 11 |- (((f C_ A /\ g C_ A) /\ (f e. Fin /\ g e. Fin) /\ (w = |^|f /\ v = |^|g)) -> E.y(y C_ A /\ y e. Fin /\ (w i^i v) = |^|y))
23		subbas.2	. . . . . . . . . . . . 13 |- B = {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
24	23	eleq2i 1961	. . . . . . . . . . . 12 |- ((w i^i v) e. B <-> (w i^i v) e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)})
25		visset 2295	. . . . . . . . . . . . . 14 |- w e. _V
26	25	inex1 3452	. . . . . . . . . . . . 13 |- (w i^i v) e. _V
27		eqeq1 1890	. . . . . . . . . . . . . . 15 |- (x = (w i^i v) -> (x = |^|y <-> (w i^i v) = |^|y))
28	27	3anbi3d 1174	. . . . . . . . . . . . . 14 |- (x = (w i^i v) -> ((y C_ A /\ y e. Fin /\ x = |^|y) <-> (y C_ A /\ y e. Fin /\ (w i^i v) = |^|y)))
29	28	exbidv 1657	. . . . . . . . . . . . 13 |- (x = (w i^i v) -> (E.y(y C_ A /\ y e. Fin /\ x = |^|y) <-> E.y(y C_ A /\ y e. Fin /\ (w i^i v) = |^|y)))
30	26, 29	elab 2403	. . . . . . . . . . . 12 |- ((w i^i v) e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} <-> E.y(y C_ A /\ y e. Fin /\ (w i^i v) = |^|y))
31	24, 30	bitr2i 191	. . . . . . . . . . 11 |- (E.y(y C_ A /\ y e. Fin /\ (w i^i v) = |^|y) <-> (w i^i v) e. B)
32	22, 31	sylib 215	. . . . . . . . . 10 |- (((f C_ A /\ g C_ A) /\ (f e. Fin /\ g e. Fin) /\ (w = |^|f /\ v = |^|g)) -> (w i^i v) e. B)
33		ssid 2634	. . . . . . . . . . 11 |- (w i^i v) C_ (w i^i v)
34	33	jctr 315	. . . . . . . . . 10 |- (u e. (w i^i v) -> (u e. (w i^i v) /\ (w i^i v) C_ (w i^i v)))
35	6, 32, 34	syl2an 503	. . . . . . . . 9 |- ((((f C_ A /\ g C_ A) /\ (f e. Fin /\ g e. Fin) /\ (w = |^|f /\ v = |^|g)) /\ u e. (w i^i v)) -> E.t e. B (u e. t /\ t C_ (w i^i v)))
36	35	ex 402	. . . . . . . 8 |- (((f C_ A /\ g C_ A) /\ (f e. Fin /\ g e. Fin) /\ (w = |^|f /\ v = |^|g)) -> (u e. (w i^i v) -> E.t e. B (u e. t /\ t C_ (w i^i v))))
37	2, 36	sylbi 216	. . . . . . 7 |- (((f C_ A /\ f e. Fin /\ w = |^|f) /\ (g C_ A /\ g e. Fin /\ v = |^|g)) -> (u e. (w i^i v) -> E.t e. B (u e. t /\ t C_ (w i^i v))))
38	37	19.23aivv 1675	. . . . . 6 |- (E.fE.g((f C_ A /\ f e. Fin /\ w = |^|f) /\ (g C_ A /\ g e. Fin /\ v = |^|g)) -> (u e. (w i^i v) -> E.t e. B (u e. t /\ t C_ (w i^i v))))
39	1, 38	sylbir 218	. . . . 5 |- ((E.f(f C_ A /\ f e. Fin /\ w = |^|f) /\ E.g(g C_ A /\ g e. Fin /\ v = |^|g)) -> (u e. (w i^i v) -> E.t e. B (u e. t /\ t C_ (w i^i v))))
40	23	eleq2i 1961	. . . . . 6 |- (w e. B <-> w e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)})
41		eqeq1 1890	. . . . . . . . 9 |- (x = w -> (x = |^|y <-> w = |^|y))
42	41	3anbi3d 1174	. . . . . . . 8 |- (x = w -> ((y C_ A /\ y e. Fin /\ x = |^|y) <-> (y C_ A /\ y e. Fin /\ w = |^|y)))
43	42	exbidv 1657	. . . . . . 7 |- (x = w -> (E.y(y C_ A /\ y e. Fin /\ x = |^|y) <-> E.y(y C_ A /\ y e. Fin /\ w = |^|y)))
44	25, 43	elab 2403	. . . . . 6 |- (w e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} <-> E.y(y C_ A /\ y e. Fin /\ w = |^|y))
45		sseq1 2637	. . . . . . . 8 |- (y = f -> (y C_ A <-> f C_ A))
46		eleq1 1957	. . . . . . . 8 |- (y = f -> (y e. Fin <-> f e. Fin))
47		inteq 3217	. . . . . . . . 9 |- (y = f -> |^|y = |^|f)
48	47	eqeq2d 1895	. . . . . . . 8 |- (y = f -> (w = |^|y <-> w = |^|f))
49	45, 46, 48	3anbi123d 1168	. . . . . . 7 |- (y = f -> ((y C_ A /\ y e. Fin /\ w = |^|y) <-> (f C_ A /\ f e. Fin /\ w = |^|f)))
50	49	cbvexv 1697	. . . . . 6 |- (E.y(y C_ A /\ y e. Fin /\ w = |^|y) <-> E.f(f C_ A /\ f e. Fin /\ w = |^|f))
51	40, 44, 50	3bitri 194	. . . . 5 |- (w e. B <-> E.f(f C_ A /\ f e. Fin /\ w = |^|f))
52	23	eleq2i 1961	. . . . . 6 |- (v e. B <-> v e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)})
53		visset 2295	. . . . . . 7 |- v e. _V
54		eqeq1 1890	. . . . . . . . 9 |- (x = v -> (x = |^|y <-> v = |^|y))
55	54	3anbi3d 1174	. . . . . . . 8 |- (x = v -> ((y C_ A /\ y e. Fin /\ x = |^|y) <-> (y C_ A /\ y e. Fin /\ v = |^|y)))
56	55	exbidv 1657	. . . . . . 7 |- (x = v -> (E.y(y C_ A /\ y e. Fin /\ x = |^|y) <-> E.y(y C_ A /\ y e. Fin /\ v = |^|y)))
57	53, 56	elab 2403	. . . . . 6 |- (v e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} <-> E.y(y C_ A /\ y e. Fin /\ v = |^|y))
58		sseq1 2637	. . . . . . . 8 |- (y = g -> (y C_ A <-> g C_ A))
59		eleq1 1957	. . . . . . . 8 |- (y = g -> (y e. Fin <-> g e. Fin))
60		inteq 3217	. . . . . . . . 9 |- (y = g -> |^|y = |^|g)
61	60	eqeq2d 1895	. . . . . . . 8 |- (y = g -> (v = |^|y <-> v = |^|g))
62	58, 59, 61	3anbi123d 1168	. . . . . . 7 |- (y = g -> ((y C_ A /\ y e. Fin /\ v = |^|y) <-> (g C_ A /\ g e. Fin /\ v = |^|g)))
63	62	cbvexv 1697	. . . . . 6 |- (E.y(y C_ A /\ y e. Fin /\ v = |^|y) <-> E.g(g C_ A /\ g e. Fin /\ v = |^|g))
64	52, 57, 63	3bitri 194	. . . . 5 |- (v e. B <-> E.g(g C_ A /\ g e. Fin /\ v = |^|g))
65	39, 51, 64	syl2anb 504	. . . 4 |- ((w e. B /\ v e. B) -> (u e. (w i^i v) -> E.t e. B (u e. t /\ t C_ (w i^i v))))
66	65	r19.21aiv 2175	. . 3 |- ((w e. B /\ v e. B) -> A.u e. (w i^i v)E.t e. B (u e. t /\ t C_ (w i^i v)))
67	66	rgen2a 2160	. 2 |- A.w e. B A.v e. B A.u e. (w i^i v)E.t e. B (u e. t /\ t C_ (w i^i v))
68		subbas.1	. . . . 5 |- A e. _V
69		df-sn 3049	. . . . . 6 |- {|^|y} = {x | x = |^|y}
70		snex 3492	. . . . . 6 |- {|^|y} e. _V
71	69, 70	eqeltrri 1968	. . . . 5 |- {x | x = |^|y} e. _V
72	68, 71	abexssex 4848	. . . 4 |- {x | E.y(y C_ A /\ x = |^|y)} e. _V
73		3simpb 873	. . . . . . 7 |- ((y C_ A /\ y e. Fin /\ x = |^|y) -> (y C_ A /\ x = |^|y))
74	73	eximi 1387	. . . . . 6 |- (E.y(y C_ A /\ y e. Fin /\ x = |^|y) -> E.y(y C_ A /\ x = |^|y))
75	74	ss2abi 2679	. . . . 5 |- {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} C_ {x | E.y(y C_ A /\ x = |^|y)}
76	23, 75	eqsstri 2647	. . . 4 |- B C_ {x | E.y(y C_ A /\ x = |^|y)}
77	72, 76	ssexi 3456	. . 3 |- B e. _V
78		isbasis2g 8881	. . 3 |- (B e. _V -> (B e. Bases <-> A.w e. B A.v e. B A.u e. (w i^i v)E.t e. B (u e. t /\ t C_ (w i^i v))))
79	77, 78	ax-mp 7	. 2 |- (B e. Bases <-> A.w e. B A.v e. B A.u e. (w i^i v)E.t e. B (u e. t /\ t C_ (w i^i v)))
80	67, 79	mpbir 207	1 |- B e. Bases

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   i^i cin 2592   C_ wss 2593  {csn 3044  |^|cint 3214  Fincfn 5426  Basesctb 8859

This theorem is referenced by:  fibas 10221  fibasOLD 15400

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-3or 859  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430  df-bases 8863

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