Theorem fctop 8920

Description: The finite complement topology on a set A. Example 3 in [Munkres] p. 77. (Contributed by FL, 15-Aug-2006.)

Hypothesis

Ref	Expression
indistop.1	|- A e. _V

Assertion

Ref	Expression
fctop	|- {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. Top

Distinct variable group:   x,A

Proof of Theorem fctop

Step	Hyp	Ref	Expression
1		indistop.1	. . . 4 |- A e. _V
2		abssexg 3490	. . . 4 |- (A e. _V -> {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. _V)
3	1, 2	ax-mp 7	. . 3 |- {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. _V
4		istopg 8865	. . 3 |- ({x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. _V -> ({x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. Top <-> (A.y(y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}) /\ A.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}A.z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} (y i^i z) e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})))
5	3, 4	ax-mp 7	. 2 |- ({x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. Top <-> (A.y(y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}) /\ A.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}A.z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} (y i^i z) e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}))
6		visset 2295	. . . . . 6 |- y e. _V
7	6	uniex 3794	. . . . 5 |- U.y e. _V
8		sseq1 2637	. . . . . 6 |- (x = U.y -> (x C_ A <-> U.y C_ A))
9		difeq2 2719	. . . . . . . 8 |- (x = U.y -> (A \ x) = (A \ U.y))
10	9	eleq1d 1963	. . . . . . 7 |- (x = U.y -> ((A \ x) e. Fin <-> (A \ U.y) e. Fin))
11		eqeq1 1890	. . . . . . 7 |- (x = U.y -> (x = (/) <-> U.y = (/)))
12	10, 11	orbi12d 689	. . . . . 6 |- (x = U.y -> (((A \ x) e. Fin \/ x = (/)) <-> ((A \ U.y) e. Fin \/ U.y = (/))))
13	8, 12	anbi12d 690	. . . . 5 |- (x = U.y -> ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) <-> (U.y C_ A /\ ((A \ U.y) e. Fin \/ U.y = (/)))))
14	7, 13	elab 2403	. . . 4 |- (U.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} <-> (U.y C_ A /\ ((A \ U.y) e. Fin \/ U.y = (/))))
15		sstr 2625	. . . . 5 |- ((U.y C_ U.{x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ U.{x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ A) -> U.y C_ A)
16		uniss 3199	. . . . 5 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y C_ U.{x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})
17		simpl 346	. . . . . . . . 9 |- ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) -> x C_ A)
18	17	a1i 8	. . . . . . . 8 |- (x e. _V -> ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) -> x C_ A))
19	18	ss2rabi 2688	. . . . . . 7 |- {x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ {x e. _V | x C_ A}
20		uniss 3199	. . . . . . 7 |- ({x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ {x e. _V | x C_ A} -> U.{x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ U.{x e. _V | x C_ A})
21	19, 20	ax-mp 7	. . . . . 6 |- U.{x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ U.{x e. _V | x C_ A}
22		rabab 2308	. . . . . . 7 |- {x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} = {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}
23	22	unieqi 3187	. . . . . 6 |- U.{x e. _V | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} = U.{x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}
24		unimax 3212	. . . . . . 7 |- (A e. _V -> U.{x e. _V | x C_ A} = A)
25	1, 24	ax-mp 7	. . . . . 6 |- U.{x e. _V | x C_ A} = A
26	21, 23, 25	3sstr3i 2655	. . . . 5 |- U.{x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} C_ A
27	15, 16, 26	sylancl 525	. . . 4 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y C_ A)
28		ssel2 2616	. . . . . . . . . . . . . . 15 |- ((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) -> z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})
29		visset 2295	. . . . . . . . . . . . . . . 16 |- z e. _V
30		sseq1 2637	. . . . . . . . . . . . . . . . 17 |- (x = z -> (x C_ A <-> z C_ A))
31		difeq2 2719	. . . . . . . . . . . . . . . . . . 19 |- (x = z -> (A \ x) = (A \ z))
32	31	eleq1d 1963	. . . . . . . . . . . . . . . . . 18 |- (x = z -> ((A \ x) e. Fin <-> (A \ z) e. Fin))
33		eqeq1 1890	. . . . . . . . . . . . . . . . . 18 |- (x = z -> (x = (/) <-> z = (/)))
34	32, 33	orbi12d 689	. . . . . . . . . . . . . . . . 17 |- (x = z -> (((A \ x) e. Fin \/ x = (/)) <-> ((A \ z) e. Fin \/ z = (/))))
35	30, 34	anbi12d 690	. . . . . . . . . . . . . . . 16 |- (x = z -> ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) <-> (z C_ A /\ ((A \ z) e. Fin \/ z = (/)))))
36	29, 35	elab 2403	. . . . . . . . . . . . . . 15 |- (z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} <-> (z C_ A /\ ((A \ z) e. Fin \/ z = (/))))
37	28, 36	sylib 215	. . . . . . . . . . . . . 14 |- ((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) -> (z C_ A /\ ((A \ z) e. Fin \/ z = (/))))
38	37	simprd 352	. . . . . . . . . . . . 13 |- ((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) -> ((A \ z) e. Fin \/ z = (/)))
39	38	ord 249	. . . . . . . . . . . 12 |- ((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) -> (-. (A \ z) e. Fin -> z = (/)))
40	39	con1d 109	. . . . . . . . . . 11 |- ((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) -> (-. z = (/) -> (A \ z) e. Fin))
41	40	imp 377	. . . . . . . . . 10 |- (((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) /\ -. z = (/)) -> (A \ z) e. Fin)
42		ssfi 5630	. . . . . . . . . . . . 13 |- (((A \ z) e. Fin /\ (A \ U.y) C_ (A \ z)) -> (A \ U.y) e. Fin)
43		elssuni 3206	. . . . . . . . . . . . . 14 |- (z e. y -> z C_ U.y)
44		sscon 2739	. . . . . . . . . . . . . 14 |- (z C_ U.y -> (A \ U.y) C_ (A \ z))
45	43, 44	syl 12	. . . . . . . . . . . . 13 |- (z e. y -> (A \ U.y) C_ (A \ z))
46	42, 45	sylan2 500	. . . . . . . . . . . 12 |- (((A \ z) e. Fin /\ z e. y) -> (A \ U.y) e. Fin)
47	46	expcom 403	. . . . . . . . . . 11 |- (z e. y -> ((A \ z) e. Fin -> (A \ U.y) e. Fin))
48	47	ad2antlr 441	. . . . . . . . . 10 |- (((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) /\ -. z = (/)) -> ((A \ z) e. Fin -> (A \ U.y) e. Fin))
49	41, 48	mpd 29	. . . . . . . . 9 |- (((y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. y) /\ -. z = (/)) -> (A \ U.y) e. Fin)
50	49	exp31 407	. . . . . . . 8 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> (z e. y -> (-. z = (/) -> (A \ U.y) e. Fin)))
51	50	r19.23adv 2215	. . . . . . 7 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> (E.z e. y -. z = (/) -> (A \ U.y) e. Fin))
52		uni0c 3204	. . . . . . . . 9 |- (U.y = (/) <-> A.z e. y z = (/))
53	52	notbii 204	. . . . . . . 8 |- (-. U.y = (/) <-> -. A.z e. y z = (/))
54		rexnal 2114	. . . . . . . 8 |- (E.z e. y -. z = (/) <-> -. A.z e. y z = (/))
55	53, 54	bitr4i 193	. . . . . . 7 |- (-. U.y = (/) <-> E.z e. y -. z = (/))
56	51, 55	syl5ib 223	. . . . . 6 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> (-. U.y = (/) -> (A \ U.y) e. Fin))
57	56	con1d 109	. . . . 5 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> (-. (A \ U.y) e. Fin -> U.y = (/)))
58	57	orrd 250	. . . 4 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> ((A \ U.y) e. Fin \/ U.y = (/)))
59	14, 27, 58	sylanbrc 527	. . 3 |- (y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})
60	59	ax-gen 1305	. 2 |- A.y(y C_ {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} -> U.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})
61		ssinss1 2821	. . . . . 6 |- (y C_ A -> (y i^i z) C_ A)
62	61	ad2antrr 440	. . . . 5 |- (((y C_ A /\ ((A \ y) e. Fin \/ y = (/))) /\ (z C_ A /\ ((A \ z) e. Fin \/ z = (/)))) -> (y i^i z) C_ A)
63		unfi 5644	. . . . . . . . 9 |- (((A \ y) e. Fin /\ (A \ z) e. Fin) -> ((A \ y) u. (A \ z)) e. Fin)
64		difindi 2849	. . . . . . . . 9 |- (A \ (y i^i z)) = ((A \ y) u. (A \ z))
65	63, 64	syl5eqel 1975	. . . . . . . 8 |- (((A \ y) e. Fin /\ (A \ z) e. Fin) -> (A \ (y i^i z)) e. Fin)
66	65	orcd 294	. . . . . . 7 |- (((A \ y) e. Fin /\ (A \ z) e. Fin) -> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))
67		ineq1 2789	. . . . . . . . 9 |- (y = (/) -> (y i^i z) = ((/) i^i z))
68		incom 2787	. . . . . . . . . 10 |- ((/) i^i z) = (z i^i (/))
69		in0 2897	. . . . . . . . . 10 |- (z i^i (/)) = (/)
70	68, 69	eqtri 1908	. . . . . . . . 9 |- ((/) i^i z) = (/)
71	67, 70	syl6eq 1944	. . . . . . . 8 |- (y = (/) -> (y i^i z) = (/))
72	71	olcd 295	. . . . . . 7 |- (y = (/) -> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))
73		ineq2 2790	. . . . . . . . 9 |- (z = (/) -> (y i^i z) = (y i^i (/)))
74		in0 2897	. . . . . . . . 9 |- (y i^i (/)) = (/)
75	73, 74	syl6eq 1944	. . . . . . . 8 |- (z = (/) -> (y i^i z) = (/))
76	75	olcd 295	. . . . . . 7 |- (z = (/) -> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))
77	66, 72, 76	ccase2 831	. . . . . 6 |- ((((A \ y) e. Fin \/ y = (/)) /\ ((A \ z) e. Fin \/ z = (/))) -> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))
78	77	ad2ant2l 444	. . . . 5 |- (((y C_ A /\ ((A \ y) e. Fin \/ y = (/))) /\ (z C_ A /\ ((A \ z) e. Fin \/ z = (/)))) -> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))
79	62, 78	jca 310	. . . 4 |- (((y C_ A /\ ((A \ y) e. Fin \/ y = (/))) /\ (z C_ A /\ ((A \ z) e. Fin \/ z = (/)))) -> ((y i^i z) C_ A /\ ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/))))
80		sseq1 2637	. . . . . . 7 |- (x = y -> (x C_ A <-> y C_ A))
81		difeq2 2719	. . . . . . . . 9 |- (x = y -> (A \ x) = (A \ y))
82	81	eleq1d 1963	. . . . . . . 8 |- (x = y -> ((A \ x) e. Fin <-> (A \ y) e. Fin))
83		eqeq1 1890	. . . . . . . 8 |- (x = y -> (x = (/) <-> y = (/)))
84	82, 83	orbi12d 689	. . . . . . 7 |- (x = y -> (((A \ x) e. Fin \/ x = (/)) <-> ((A \ y) e. Fin \/ y = (/))))
85	80, 84	anbi12d 690	. . . . . 6 |- (x = y -> ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) <-> (y C_ A /\ ((A \ y) e. Fin \/ y = (/)))))
86	6, 85	elab 2403	. . . . 5 |- (y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} <-> (y C_ A /\ ((A \ y) e. Fin \/ y = (/))))
87	86, 36	anbi12i 540	. . . 4 |- ((y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}) <-> ((y C_ A /\ ((A \ y) e. Fin \/ y = (/))) /\ (z C_ A /\ ((A \ z) e. Fin \/ z = (/)))))
88	6	inex1 3452	. . . . 5 |- (y i^i z) e. _V
89		sseq1 2637	. . . . . 6 |- (x = (y i^i z) -> (x C_ A <-> (y i^i z) C_ A))
90		difeq2 2719	. . . . . . . 8 |- (x = (y i^i z) -> (A \ x) = (A \ (y i^i z)))
91	90	eleq1d 1963	. . . . . . 7 |- (x = (y i^i z) -> ((A \ x) e. Fin <-> (A \ (y i^i z)) e. Fin))
92		eqeq1 1890	. . . . . . 7 |- (x = (y i^i z) -> (x = (/) <-> (y i^i z) = (/)))
93	91, 92	orbi12d 689	. . . . . 6 |- (x = (y i^i z) -> (((A \ x) e. Fin \/ x = (/)) <-> ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/))))
94	89, 93	anbi12d 690	. . . . 5 |- (x = (y i^i z) -> ((x C_ A /\ ((A \ x) e. Fin \/ x = (/))) <-> ((y i^i z) C_ A /\ ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/)))))
95	88, 94	elab 2403	. . . 4 |- ((y i^i z) e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} <-> ((y i^i z) C_ A /\ ((A \ (y i^i z)) e. Fin \/ (y i^i z) = (/))))
96	79, 87, 95	3imtr4i 236	. . 3 |- ((y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} /\ z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}) -> (y i^i z) e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))})
97	96	rgen2a 2160	. 2 |- A.y e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}A.z e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} (y i^i z) e. {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))}
98	5, 60, 97	mpbir2an 800	1 |- {x | (x C_ A /\ ((A \ x) e. Fin \/ x = (/)))} e. Top

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  Fincfn 5426  Topctop 8857

This theorem is referenced by:  fctop2 8921

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-top 8861

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