Theorem elcls3 7831

Description: Membership in a closure in terms of the members of a basis. Theorem 6.5(b) of [Munkres] p. 95.

Hypotheses

Ref	Expression
elcls3.1	|- J = (topGen` B)
elcls3.2	|- X = U.J

Assertion

Ref	Expression
elcls3	|- ((B e. Bases /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. B (P e. x -> (x i^i S) =/= (/))))

Distinct variable groups:   x,B   x,J   x,P   x,S   x,X

Proof of Theorem elcls3

Step	Hyp	Ref	Expression
1		elcls3.2	. . . 4 |- X = U.J
2	1	elcls 7824	. . 3 |- ((J e. Top /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. J (P e. x -> (x i^i S) =/= (/))))
3		tgcl 7748	. . . 4 |- (B e. Bases -> (topGen` B) e. Top)
4		elcls3.1	. . . 4 |- J = (topGen` B)
5	3, 4	syl5eqel 1589	. . 3 |- (B e. Bases -> J e. Top)
6	2, 5	syl3an1 862	. 2 |- ((B e. Bases /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. J (P e. x -> (x i^i S) =/= (/))))
7		bastg 7746	. . . . . . . 8 |- (B e. Bases -> B (_ (topGen` B))
8	7, 4	syl6ssr 2152	. . . . . . 7 |- (B e. Bases -> B (_ J)
9	8	sseld 2111	. . . . . 6 |- (B e. Bases -> (x e. B -> x e. J))
10	9	imim1d 28	. . . . 5 |- (B e. Bases -> ((x e. J -> (P e. x -> (x i^i S) =/= (/))) -> (x e. B -> (P e. x -> (x i^i S) =/= (/)))))
11	10	r19.20dv2 1749	. . . 4 |- (B e. Bases -> (A.x e. J (P e. x -> (x i^i S) =/= (/)) -> A.x e. B (P e. x -> (x i^i S) =/= (/))))
12	11	3ad2ant1 803	. . 3 |- ((B e. Bases /\ S (_ X /\ P e. X) -> (A.x e. J (P e. x -> (x i^i S) =/= (/)) -> A.x e. B (P e. x -> (x i^i S) =/= (/))))
13		tg2 7745	. . . . . . . . . . 11 |- ((B e. Bases /\ x e. (topGen` B) /\ P e. x) -> E.z e. B (P e. z /\ z (_ x))
14	4	eleq2i 1575	. . . . . . . . . . 11 |- (x e. J <-> x e. (topGen` B))
15	13, 14	syl3an2b 866	. . . . . . . . . 10 |- ((B e. Bases /\ x e. J /\ P e. x) -> E.z e. B (P e. z /\ z (_ x))
16	15	3expb 837	. . . . . . . . 9 |- ((B e. Bases /\ (x e. J /\ P e. x)) -> E.z e. B (P e. z /\ z (_ x))
17	16	adantlr 393	. . . . . . . 8 |- (((B e. Bases /\ A.y e. B (P e. y -> (y i^i S) =/= (/))) /\ (x e. J /\ P e. x)) -> E.z e. B (P e. z /\ z (_ x))
18		ssdisj 2363	. . . . . . . . . . . . . . 15 |- ((z (_ x /\ (x i^i S) = (/)) -> (z i^i S) = (/))
19	18	ex 371	. . . . . . . . . . . . . 14 |- (z (_ x -> ((x i^i S) = (/) -> (z i^i S) = (/)))
20	19	necon3d 1641	. . . . . . . . . . . . 13 |- (z (_ x -> ((z i^i S) =/= (/) -> (x i^i S) =/= (/)))
21		eleq2 1572	. . . . . . . . . . . . . . . 16 |- (y = z -> (P e. y <-> P e. z))
22		ineq1 2254	. . . . . . . . . . . . . . . . 17 |- (y = z -> (y i^i S) = (z i^i S))
23	22	neeq1d 1631	. . . . . . . . . . . . . . . 16 |- (y = z -> ((y i^i S) =/= (/) <-> (z i^i S) =/= (/)))
24	21, 23	imbi12d 628	. . . . . . . . . . . . . . 15 |- (y = z -> ((P e. y -> (y i^i S) =/= (/)) <-> (P e. z -> (z i^i S) =/= (/))))
25	24	rcla4cva 1914	. . . . . . . . . . . . . 14 |- ((A.y e. B (P e. y -> (y i^i S) =/= (/)) /\ z e. B) -> (P e. z -> (z i^i S) =/= (/)))
26	25	imp 348	. . . . . . . . . . . . 13 |- (((A.y e. B (P e. y -> (y i^i S) =/= (/)) /\ z e. B) /\ P e. z) -> (z i^i S) =/= (/))
27	20, 26	syl5com 52	. . . . . . . . . . . 12 |- (((A.y e. B (P e. y -> (y i^i S) =/= (/)) /\ z e. B) /\ P e. z) -> (z (_ x -> (x i^i S) =/= (/)))
28	27	exp31 376	. . . . . . . . . . 11 |- (A.y e. B (P e. y -> (y i^i S) =/= (/)) -> (z e. B -> (P e. z -> (z (_ x -> (x i^i S) =/= (/)))))
29	28	imp4a 362	. . . . . . . . . 10 |- (A.y e. B (P e. y -> (y i^i S) =/= (/)) -> (z e. B -> ((P e. z /\ z (_ x) -> (x i^i S) =/= (/))))
30	29	r19.23adv 1784	. . . . . . . . 9 |- (A.y e. B (P e. y -> (y i^i S) =/= (/)) -> (E.z e. B (P e. z /\ z (_ x) -> (x i^i S) =/= (/)))
31	30	ad2antlr 405	. . . . . . . 8 |- (((B e. Bases /\ A.y e. B (P e. y -> (y i^i S) =/= (/))) /\ (x e. J /\ P e. x)) -> (E.z e. B (P e. z /\ z (_ x) -> (x i^i S) =/= (/)))
32	17, 31	mpd 26	. . . . . . 7 |- (((B e. Bases /\ A.y e. B (P e. y -> (y i^i S) =/= (/))) /\ (x e. J /\ P e. x)) -> (x i^i S) =/= (/))
33	32	exp43 384	. . . . . 6 |- (B e. Bases -> (A.y e. B (P e. y -> (y i^i S) =/= (/)) -> (x e. J -> (P e. x -> (x i^i S) =/= (/)))))
34	33	r19.21adv 1756	. . . . 5 |- (B e. Bases -> (A.y e. B (P e. y -> (y i^i S) =/= (/)) -> A.x e. J (P e. x -> (x i^i S) =/= (/))))
35		eleq2 1572	. . . . . . 7 |- (x = y -> (P e. x <-> P e. y))
36		ineq1 2254	. . . . . . . 8 |- (x = y -> (x i^i S) = (y i^i S))
37	36	neeq1d 1631	. . . . . . 7 |- (x = y -> ((x i^i S) =/= (/) <-> (y i^i S) =/= (/)))
38	35, 37	imbi12d 628	. . . . . 6 |- (x = y -> ((P e. x -> (x i^i S) =/= (/)) <-> (P e. y -> (y i^i S) =/= (/))))
39	38	cbvralv 1838	. . . . 5 |- (A.x e. B (P e. x -> (x i^i S) =/= (/)) <-> A.y e. B (P e. y -> (y i^i S) =/= (/)))
40	34, 39	syl5ib 204	. . . 4 |- (B e. Bases -> (A.x e. B (P e. x -> (x i^i S) =/= (/)) -> A.x e. J (P e. x -> (x i^i S) =/= (/))))
41	40	3ad2ant1 803	. . 3 |- ((B e. Bases /\ S (_ X /\ P e. X) -> (A.x e. B (P e. x -> (x i^i S) =/= (/)) -> A.x e. J (P e. x -> (x i^i S) =/= (/))))
42	12, 41	impbid 518	. 2 |- ((B e. Bases /\ S (_ X /\ P e. X) -> (A.x e. J (P e. x -> (x i^i S) =/= (/)) <-> A.x e. B (P e. x -> (x i^i S) =/= (/))))
43	6, 42	bitrd 530	1 |- ((B e. Bases /\ S (_ X /\ P e. X) -> (P e. ((cls` J)` S) <-> A.x e. B (P e. x -> (x i^i S) =/= (/))))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   /\ w3a 778   = wceq 988   e. wcel 990   =/= wne 1622  A.wral 1683  E.wrex 1684   i^i cin 2090   (_ wss 2091  (/)c0 2324  U.cuni 2551  ` cfv 3237  Topctop 7713  Basesctb 7715  topGenctg 7716  clsccl 7782

This theorem is referenced by:  qdensere 7871

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-rab 1690  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-int 2582  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253  df-top 7717  df-bases 7719  df-topgen 7720  df-cld 7783  df-ntr 7784  df-cls 7785

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