Theorem subbas2 8915

Description: The collection of finite intersections of any set (subbasis) A is a basis for a topology on U.A. Justifies the definition of subbasis in [Munkres] p. 82 (after using unitg 8893).

Hypothesis

Ref	Expression
subbas2.1	|- B = {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}

Assertion

Ref	Expression
subbas2	|- U.B = U.A

Distinct variable group:   x,y,A

Proof of Theorem subbas2

Step	Hyp	Ref	Expression
1		subbas2.1	. . 3 |- B = {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
2	1	unieqi 3187	. 2 |- U.B = U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
3		eluniab 3189	. . . . 5 |- (w e. U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} <-> E.x(w e. x /\ E.y(y C_ A /\ y e. Fin /\ x = |^|y)))
4		ssel 2615	. . . . . . . . . 10 |- (x C_ U.A -> (w e. x -> w e. U.A))
5	4	com12 14	. . . . . . . . 9 |- (w e. x -> (x C_ U.A -> w e. U.A))
6		simpr 350	. . . . . . . . . . 11 |- ((y C_ A /\ x = |^|y) -> x = |^|y)
7		intssuni2 3243	. . . . . . . . . . . 12 |- ((y C_ A /\ y =/= (/)) -> |^|y C_ U.A)
8		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
9		eleq1 1957	. . . . . . . . . . . . . 14 |- (x = |^|y -> (x e. _V <-> |^|y e. _V))
10	8, 9	mpbii 210	. . . . . . . . . . . . 13 |- (x = |^|y -> |^|y e. _V)
11		intex 3465	. . . . . . . . . . . . 13 |- (y =/= (/) <-> |^|y e. _V)
12	10, 11	sylibr 217	. . . . . . . . . . . 12 |- (x = |^|y -> y =/= (/))
13	7, 12	sylan2 500	. . . . . . . . . . 11 |- ((y C_ A /\ x = |^|y) -> |^|y C_ U.A)
14	6, 13	eqsstrd 2651	. . . . . . . . . 10 |- ((y C_ A /\ x = |^|y) -> x C_ U.A)
15	14	3adant2 895	. . . . . . . . 9 |- ((y C_ A /\ y e. Fin /\ x = |^|y) -> x C_ U.A)
16	5, 15	syl5 20	. . . . . . . 8 |- (w e. x -> ((y C_ A /\ y e. Fin /\ x = |^|y) -> w e. U.A))
17	16	19.23adv 1584	. . . . . . 7 |- (w e. x -> (E.y(y C_ A /\ y e. Fin /\ x = |^|y) -> w e. U.A))
18	17	imp 377	. . . . . 6 |- ((w e. x /\ E.y(y C_ A /\ y e. Fin /\ x = |^|y)) -> w e. U.A)
19	18	19.23aiv 1674	. . . . 5 |- (E.x(w e. x /\ E.y(y C_ A /\ y e. Fin /\ x = |^|y)) -> w e. U.A)
20	3, 19	sylbi 216	. . . 4 |- (w e. U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} -> w e. U.A)
21	20	ssriv 2621	. . 3 |- U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} C_ U.A
22		snssi 3129	. . . . . . 7 |- (w e. A -> {w} C_ A)
23		snfi 5491	. . . . . . . 8 |- {w} e. Fin
24	23	a1i 8	. . . . . . 7 |- (w e. A -> {w} e. Fin)
25		visset 2295	. . . . . . . . . 10 |- w e. _V
26	25	intsn 3252	. . . . . . . . 9 |- |^|{w} = w
27	26	eqcomi 1888	. . . . . . . 8 |- w = |^|{w}
28	27	a1i 8	. . . . . . 7 |- (w e. A -> w = |^|{w})
29		snex 3492	. . . . . . . 8 |- {w} e. _V
30		sseq1 2637	. . . . . . . . 9 |- (y = {w} -> (y C_ A <-> {w} C_ A))
31		eleq1 1957	. . . . . . . . 9 |- (y = {w} -> (y e. Fin <-> {w} e. Fin))
32		inteq 3217	. . . . . . . . . 10 |- (y = {w} -> |^|y = |^|{w})
33	32	eqeq2d 1895	. . . . . . . . 9 |- (y = {w} -> (w = |^|y <-> w = |^|{w}))
34	30, 31, 33	3anbi123d 1168	. . . . . . . 8 |- (y = {w} -> ((y C_ A /\ y e. Fin /\ w = |^|y) <-> ({w} C_ A /\ {w} e. Fin /\ w = |^|{w})))
35	29, 34	cla4ev 2371	. . . . . . 7 |- (({w} C_ A /\ {w} e. Fin /\ w = |^|{w}) -> E.y(y C_ A /\ y e. Fin /\ w = |^|y))
36	22, 24, 28, 35	syl111anc 1100	. . . . . 6 |- (w e. A -> E.y(y C_ A /\ y e. Fin /\ w = |^|y))
37		eqeq1 1890	. . . . . . . . 9 |- (x = w -> (x = |^|y <-> w = |^|y))
38	37	3anbi3d 1174	. . . . . . . 8 |- (x = w -> ((y C_ A /\ y e. Fin /\ x = |^|y) <-> (y C_ A /\ y e. Fin /\ w = |^|y)))
39	38	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)))
40	25, 39	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))
41	36, 40	sylibr 217	. . . . 5 |- (w e. A -> w e. {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)})
42	41	ssriv 2621	. . . 4 |- A C_ {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
43		uniss 3199	. . . 4 |- (A C_ {x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} -> U.A C_ U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)})
44	42, 43	ax-mp 7	. . 3 |- U.A C_ U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)}
45	21, 44	eqssi 2632	. 2 |- U.{x | E.y(y C_ A /\ y e. Fin /\ x = |^|y)} = U.A
46	2, 45	eqtri 1908	1 |- U.B = U.A

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  |^|cint 3214  Fincfn 5426

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-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-rab 2112  df-v 2294  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-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-1o 5177  df-en 5427  df-fin 5430

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