Theorem elcls3 7708

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 7701	. . 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		tgclt 7623	. . . 4 |- (B e. Bases -> (topGen` B) e. Top)
4		elcls3.1	. . . 4 |- J = (topGen` B)
5	3, 4	syl5eqel 1555	. . 3 |- (B e. Bases -> J e. Top)
6	2, 5	syl3an1 861	. 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		bastgt 7621	. . . . . . . 8 |- (B e. Bases -> B (_ (topGen` B))
8	7, 4	syl6ssr 2111	. . . . . . 7 |- (B e. Bases -> B (_ J)
9	8	sseld 2070	. . . . . 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 1714	. . . 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 802	. . 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		tg2t 7620	. . . . . . . . . . 11 |- ((B e. Bases /\ x e. (topGen` B) /\ P e. x) -> E.z e. B (P e. z /\ z (_ x))
14	4	eleq2i 1541	. . . . . . . . . . 11 |- (x e. J <-> x e. (topGen` B))
15	13, 14	syl3an2b 865	. . . . . . . . . 10 |- ((B e. Bases /\ x e. J /\ P e. x) -> E.z e. B (P e. z /\ z (_ x))
16	15	3expb 836	. . . . . . . . 9 |- ((B e. Bases /\ (x e. J /\ P e. x)) -> E.z e. B (P e. z /\ z (_ x))
17	16	adantlr 395	. . . . . . . 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 2322	. . . . . . . . . . . . . . 15 |- ((z (_ x /\ (x i^i S) = (/)) -> (z i^i S) = (/))
19	18	ex 373	. . . . . . . . . . . . . 14 |- (z (_ x -> ((x i^i S) = (/) -> (z i^i S) = (/)))
20	19	necon3d 1607	. . . . . . . . . . . . 13 |- (z (_ x -> ((z i^i S) =/= (/) -> (x i^i S) =/= (/)))
21		eleq2 1538	. . . . . . . . . . . . . . . 16 |- (y = z -> (P e. y <-> P e. z))
22		ineq1 2213	. . . . . . . . . . . . . . . . 17 |- (y = z -> (y i^i S) = (z i^i S))
23	22	neeq1d 1597	. . . . . . . . . . . . . . . 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 1879	. . . . . . . . . . . . . 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 350	. . . . . . . . . . . . 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 378	. . . . . . . . . . 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 364	. . . . . . . . . 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 1749	. . . . . . . . 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 407	. . . . . . . 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 386	. . . . . 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 1721	. . . . 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 1538	. . . . . . 7 |- (x = y -> (P e. x <-> P e. y))
36		ineq1 2213	. . . . . . . 8 |- (x = y -> (x i^i S) = (y i^i S))
37	36	neeq1d 1597	. . . . . . 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 1803	. . . . 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 206	. . . 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 802	. . 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 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   =/= wne 1588  A.wral 1648  E.wrex 1649   i^i cin 2049   (_ wss 2050  (/)c0 2283  U.cuni 2507  ` cfv 3188  Topctop 7590  Basesctb 7592  topGenctg 7593  clsccl 7659

This theorem is referenced by:  qdensere 7748

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-top 7594  df-bases 7596  df-topgen 7597  df-cld 7660  df-ntr 7661  df-cls 7662

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