Theorem filrn 10293

Description: Given a filter on a domain, produce a filter on the range. (Contributed by Jeff Hankins, 7-Sep-2009.)

Hypotheses

Ref	Expression
filrn.1	|- X = U.B
filrn.2	|- C = {y | E.x e. B y = (F"x)}

Assertion

Ref	Expression
filrn	|- ((B e. fBas /\ F Fn X) -> C e. fBas)

Distinct variable groups:   x,y,B   x,F,y   x,X

Proof of Theorem filrn

Step	Hyp	Ref	Expression
1		funimaexg 4495	. . . . . . . . . . 11 |- ((Fun F /\ b e. B) -> (F"b) e. _V)
2		fnfun 4510	. . . . . . . . . . 11 |- (F Fn X -> Fun F)
3	1, 2	sylan 497	. . . . . . . . . 10 |- ((F Fn X /\ b e. B) -> (F"b) e. _V)
4	3	ancoms 484	. . . . . . . . 9 |- ((b e. B /\ F Fn X) -> (F"b) e. _V)
5		eqid 1884	. . . . . . . . . . 11 |- (F"b) = (F"b)
6		imaeq2 4260	. . . . . . . . . . . . 13 |- (x = b -> (F"x) = (F"b))
7	6	eqeq2d 1895	. . . . . . . . . . . 12 |- (x = b -> ((F"b) = (F"x) <-> (F"b) = (F"b)))
8	7	rcla4ev 2381	. . . . . . . . . . 11 |- ((b e. B /\ (F"b) = (F"b)) -> E.x e. B (F"b) = (F"x))
9	5, 8	mpan2 760	. . . . . . . . . 10 |- (b e. B -> E.x e. B (F"b) = (F"x))
10	9	adantr 425	. . . . . . . . 9 |- ((b e. B /\ F Fn X) -> E.x e. B (F"b) = (F"x))
11		eqeq1 1890	. . . . . . . . . . 11 |- (c = (F"b) -> (c = (F"x) <-> (F"b) = (F"x)))
12	11	rexbidv 2124	. . . . . . . . . 10 |- (c = (F"b) -> (E.x e. B c = (F"x) <-> E.x e. B (F"b) = (F"x)))
13	12	cla4egv 2365	. . . . . . . . 9 |- ((F"b) e. _V -> (E.x e. B (F"b) = (F"x) -> E.cE.x e. B c = (F"x)))
14	4, 10, 13	sylc 83	. . . . . . . 8 |- ((b e. B /\ F Fn X) -> E.cE.x e. B c = (F"x))
15	14	ex 402	. . . . . . 7 |- (b e. B -> (F Fn X -> E.cE.x e. B c = (F"x)))
16		filrn.2	. . . . . . . . . . 11 |- C = {y | E.x e. B y = (F"x)}
17	16	eleq2i 1961	. . . . . . . . . 10 |- (c e. C <-> c e. {y | E.x e. B y = (F"x)})
18		visset 2295	. . . . . . . . . . 11 |- c e. _V
19		eqeq1 1890	. . . . . . . . . . . 12 |- (y = c -> (y = (F"x) <-> c = (F"x)))
20	19	rexbidv 2124	. . . . . . . . . . 11 |- (y = c -> (E.x e. B y = (F"x) <-> E.x e. B c = (F"x)))
21	18, 20	elab 2403	. . . . . . . . . 10 |- (c e. {y | E.x e. B y = (F"x)} <-> E.x e. B c = (F"x))
22	17, 21	bitr2i 191	. . . . . . . . 9 |- (E.x e. B c = (F"x) <-> c e. C)
23	22	exbii 1398	. . . . . . . 8 |- (E.cE.x e. B c = (F"x) <-> E.c c e. C)
24		n0 2884	. . . . . . . 8 |- (C =/= (/) <-> E.c c e. C)
25	23, 24	bitr4i 193	. . . . . . 7 |- (E.cE.x e. B c = (F"x) <-> C =/= (/))
26	15, 25	syl6ib 229	. . . . . 6 |- (b e. B -> (F Fn X -> C =/= (/)))
27	26	19.23aiv 1674	. . . . 5 |- (E.b b e. B -> (F Fn X -> C =/= (/)))
28	27	imp 377	. . . 4 |- ((E.b b e. B /\ F Fn X) -> C =/= (/))
29		fbasne0 10262	. . . . 5 |- (B e. fBas -> B =/= (/))
30		n0 2884	. . . . 5 |- (B =/= (/) <-> E.b b e. B)
31	29, 30	sylib 215	. . . 4 |- (B e. fBas -> E.b b e. B)
32	28, 31	sylan 497	. . 3 |- ((B e. fBas /\ F Fn X) -> C =/= (/))
33		elssuni 3206	. . . . . . . 8 |- (x e. B -> x C_ U.B)
34	33	a1i 8	. . . . . . 7 |- ((B e. fBas /\ F Fn X) -> (x e. B -> x C_ U.B))
35		eleq1 1957	. . . . . . . . . . 11 |- (x = (/) -> (x e. B <-> (/) e. B))
36	35	notbid 673	. . . . . . . . . 10 |- (x = (/) -> (-. x e. B <-> -. (/) e. B))
37		0nelfb 10277	. . . . . . . . . . 11 |- (B e. fBas -> (/) e/ B)
38		df-nel 2020	. . . . . . . . . . 11 |- ((/) e/ B <-> -. (/) e. B)
39	37, 38	sylib 215	. . . . . . . . . 10 |- (B e. fBas -> -. (/) e. B)
40	36, 39	syl5cbir 228	. . . . . . . . 9 |- (B e. fBas -> (x = (/) -> -. x e. B))
41	40	con2d 107	. . . . . . . 8 |- (B e. fBas -> (x e. B -> -. x = (/)))
42	41	adantr 425	. . . . . . 7 |- ((B e. fBas /\ F Fn X) -> (x e. B -> -. x = (/)))
43	34, 42	jcad 661	. . . . . 6 |- ((B e. fBas /\ F Fn X) -> (x e. B -> (x C_ U.B /\ -. x = (/))))
44		fndm 4512	. . . . . . . . . . . . . . . 16 |- (F Fn X -> dom F = X)
45	44	adantl 424	. . . . . . . . . . . . . . 15 |- ((B e. fBas /\ F Fn X) -> dom F = X)
46		filrn.1	. . . . . . . . . . . . . . 15 |- X = U.B
47	45, 46	syl6eq 1944	. . . . . . . . . . . . . 14 |- ((B e. fBas /\ F Fn X) -> dom F = U.B)
48	47	sseq2d 2645	. . . . . . . . . . . . 13 |- ((B e. fBas /\ F Fn X) -> (x C_ dom F <-> x C_ U.B))
49	48	biimpar 461	. . . . . . . . . . . 12 |- (((B e. fBas /\ F Fn X) /\ x C_ U.B) -> x C_ dom F)
50		sseqin2 2811	. . . . . . . . . . . 12 |- (x C_ dom F <-> (dom F i^i x) = x)
51	49, 50	sylib 215	. . . . . . . . . . 11 |- (((B e. fBas /\ F Fn X) /\ x C_ U.B) -> (dom F i^i x) = x)
52	51	eqeq1d 1892	. . . . . . . . . 10 |- (((B e. fBas /\ F Fn X) /\ x C_ U.B) -> ((dom F i^i x) = (/) <-> x = (/)))
53	52	biimpd 170	. . . . . . . . 9 |- (((B e. fBas /\ F Fn X) /\ x C_ U.B) -> ((dom F i^i x) = (/) -> x = (/)))
54	53	con3d 111	. . . . . . . 8 |- (((B e. fBas /\ F Fn X) /\ x C_ U.B) -> (-. x = (/) -> -. (dom F i^i x) = (/)))
55	54	expimpd 404	. . . . . . 7 |- ((B e. fBas /\ F Fn X) -> ((x C_ U.B /\ -. x = (/)) -> -. (dom F i^i x) = (/)))
56		eqcom 1886	. . . . . . . . 9 |- ((/) = (F"x) <-> (F"x) = (/))
57		imadisj 4285	. . . . . . . . 9 |- ((F"x) = (/) <-> (dom F i^i x) = (/))
58	56, 57	bitri 190	. . . . . . . 8 |- ((/) = (F"x) <-> (dom F i^i x) = (/))
59	58	notbii 204	. . . . . . 7 |- (-. (/) = (F"x) <-> -. (dom F i^i x) = (/))
60	55, 59	syl6ibr 230	. . . . . 6 |- ((B e. fBas /\ F Fn X) -> ((x C_ U.B /\ -. x = (/)) -> -. (/) = (F"x)))
61	43, 60	syld 30	. . . . 5 |- ((B e. fBas /\ F Fn X) -> (x e. B -> -. (/) = (F"x)))
62	61	r19.21aiv 2175	. . . 4 |- ((B e. fBas /\ F Fn X) -> A.x e. B -. (/) = (F"x))
63	16	eleq2i 1961	. . . . . . 7 |- ((/) e. C <-> (/) e. {y | E.x e. B y = (F"x)})
64		0ex 3446	. . . . . . . 8 |- (/) e. _V
65		eqeq1 1890	. . . . . . . . 9 |- (y = (/) -> (y = (F"x) <-> (/) = (F"x)))
66	65	rexbidv 2124	. . . . . . . 8 |- (y = (/) -> (E.x e. B y = (F"x) <-> E.x e. B (/) = (F"x)))
67	64, 66	elab 2403	. . . . . . 7 |- ((/) e. {y | E.x e. B y = (F"x)} <-> E.x e. B (/) = (F"x))
68	63, 67	bitr2i 191	. . . . . 6 |- (E.x e. B (/) = (F"x) <-> (/) e. C)
69	68	notbii 204	. . . . 5 |- (-. E.x e. B (/) = (F"x) <-> -. (/) e. C)
70		ralnex 2113	. . . . 5 |- (A.x e. B -. (/) = (F"x) <-> -. E.x e. B (/) = (F"x))
71		df-nel 2020	. . . . 5 |- ((/) e/ C <-> -. (/) e. C)
72	69, 70, 71	3bitr4i 200	. . . 4 |- (A.x e. B -. (/) = (F"x) <-> (/) e/ C)
73	62, 72	sylib 215	. . 3 |- ((B e. fBas /\ F Fn X) -> (/) e/ C)
74		fbasssin 10278	. . . . . . . . . 10 |- ((B e. fBas /\ u e. B /\ v e. B) -> E.b e. B b C_ (u i^i v))
75	74	3expb 1068	. . . . . . . . 9 |- ((B e. fBas /\ (u e. B /\ v e. B)) -> E.b e. B b C_ (u i^i v))
76	75	ad2ant2r 445	. . . . . . . 8 |- (((B e. fBas /\ F Fn X) /\ ((u e. B /\ v e. B) /\ (r = (F"u) /\ s = (F"v)))) -> E.b e. B b C_ (u i^i v))
77	9	ad2antrl 442	. . . . . . . . . . . . 13 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> E.x e. B (F"b) = (F"x))
78		visset 2295	. . . . . . . . . . . . . . . . . 18 |- b e. _V
79	78	funimaex 4496	. . . . . . . . . . . . . . . . 17 |- (Fun F -> (F"b) e. _V)
80		eqeq1 1890	. . . . . . . . . . . . . . . . . . 19 |- (y = (F"b) -> (y = (F"x) <-> (F"b) = (F"x)))
81	80	rexbidv 2124	. . . . . . . . . . . . . . . . . 18 |- (y = (F"b) -> (E.x e. B y = (F"x) <-> E.x e. B (F"b) = (F"x)))
82	81	elabg 2405	. . . . . . . . . . . . . . . . 17 |- ((F"b) e. _V -> ((F"b) e. {y | E.x e. B y = (F"x)} <-> E.x e. B (F"b) = (F"x)))
83	2, 79, 82	3syl 24	. . . . . . . . . . . . . . . 16 |- (F Fn X -> ((F"b) e. {y | E.x e. B y = (F"x)} <-> E.x e. B (F"b) = (F"x)))
84	16	eleq2i 1961	. . . . . . . . . . . . . . . 16 |- ((F"b) e. C <-> (F"b) e. {y | E.x e. B y = (F"x)})
85	83, 84	syl5bb 591	. . . . . . . . . . . . . . 15 |- (F Fn X -> ((F"b) e. C <-> E.x e. B (F"b) = (F"x)))
86	85	adantl 424	. . . . . . . . . . . . . 14 |- ((B e. fBas /\ F Fn X) -> ((F"b) e. C <-> E.x e. B (F"b) = (F"x)))
87	86	ad2antrr 440	. . . . . . . . . . . . 13 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> ((F"b) e. C <-> E.x e. B (F"b) = (F"x)))
88	77, 87	mpbird 213	. . . . . . . . . . . 12 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (F"b) e. C)
89		imass2 4299	. . . . . . . . . . . . . 14 |- (b C_ (u i^i v) -> (F"b) C_ (F"(u i^i v)))
90	89	ad2antll 443	. . . . . . . . . . . . 13 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (F"b) C_ (F"(u i^i v)))
91		inss1 2812	. . . . . . . . . . . . . . . . 17 |- (u i^i v) C_ u
92		imass2 4299	. . . . . . . . . . . . . . . . 17 |- ((u i^i v) C_ u -> (F"(u i^i v)) C_ (F"u))
93	91, 92	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (F"(u i^i v)) C_ (F"u)
94		inss2 2813	. . . . . . . . . . . . . . . . 17 |- (u i^i v) C_ v
95		imass2 4299	. . . . . . . . . . . . . . . . 17 |- ((u i^i v) C_ v -> (F"(u i^i v)) C_ (F"v))
96	94, 95	ax-mp 7	. . . . . . . . . . . . . . . 16 |- (F"(u i^i v)) C_ (F"v)
97	93, 96	ssini 2816	. . . . . . . . . . . . . . 15 |- (F"(u i^i v)) C_ ((F"u) i^i (F"v))
98	97	a1i 8	. . . . . . . . . . . . . 14 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (F"(u i^i v)) C_ ((F"u) i^i (F"v)))
99		ineq12 2791	. . . . . . . . . . . . . . 15 |- ((r = (F"u) /\ s = (F"v)) -> (r i^i s) = ((F"u) i^i (F"v)))
100	99	ad2antlr 441	. . . . . . . . . . . . . 14 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (r i^i s) = ((F"u) i^i (F"v)))
101	98, 100	sseqtr4d 2654	. . . . . . . . . . . . 13 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (F"(u i^i v)) C_ (r i^i s))
102	90, 101	sstrd 2627	. . . . . . . . . . . 12 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> (F"b) C_ (r i^i s))
103		sseq1 2637	. . . . . . . . . . . . 13 |- (t = (F"b) -> (t C_ (r i^i s) <-> (F"b) C_ (r i^i s)))
104	103	rcla4ev 2381	. . . . . . . . . . . 12 |- (((F"b) e. C /\ (F"b) C_ (r i^i s)) -> E.t e. C t C_ (r i^i s))
105	88, 102, 104	syl11anc 524	. . . . . . . . . . 11 |- ((((B e. fBas /\ F Fn X) /\ (r = (F"u) /\ s = (F"v))) /\ (b e. B /\ b C_ (u i^i v))) -> E.t e. C t C_ (r i^i s))
106	105	adantlrl 434	. . . . . . . . . 10 |- ((((B e. fBas /\ F Fn X) /\ ((u e. B /\ v e. B) /\ (r = (F"u) /\ s = (F"v)))) /\ (b e. B /\ b C_ (u i^i v))) -> E.t e. C t C_ (r i^i s))
107	106	exp32 408	. . . . . . . . 9 |- (((B e. fBas /\ F Fn X) /\ ((u e. B /\ v e. B) /\ (r = (F"u) /\ s = (F"v)))) -> (b e. B -> (b C_ (u i^i v) -> E.t e. C t C_ (r i^i s))))
108	107	r19.23adv 2215	. . . . . . . 8 |- (((B e. fBas /\ F Fn X) /\ ((u e. B /\ v e. B) /\ (r = (F"u) /\ s = (F"v)))) -> (E.b e. B b C_ (u i^i v) -> E.t e. C t C_ (r i^i s)))
109	76, 108	mpd 29	. . . . . . 7 |- (((B e. fBas /\ F Fn X) /\ ((u e. B /\ v e. B) /\ (r = (F"u) /\ s = (F"v)))) -> E.t e. C t C_ (r i^i s))
110	109	exp32 408	. . . . . 6 |- ((B e. fBas /\ F Fn X) -> ((u e. B /\ v e. B) -> ((r = (F"u) /\ s = (F"v)) -> E.t e. C t C_ (r i^i s))))
111	110	r19.23advv 2218	. . . . 5 |- ((B e. fBas /\ F Fn X) -> (E.u e. B E.v e. B (r = (F"u) /\ s = (F"v)) -> E.t e. C t C_ (r i^i s)))
112	16	eleq2i 1961	. . . . . . . 8 |- (r e. C <-> r e. {y | E.x e. B y = (F"x)})
113		visset 2295	. . . . . . . . 9 |- r e. _V
114		eqeq1 1890	. . . . . . . . . 10 |- (y = r -> (y = (F"x) <-> r = (F"x)))
115	114	rexbidv 2124	. . . . . . . . 9 |- (y = r -> (E.x e. B y = (F"x) <-> E.x e. B r = (F"x)))
116	113, 115	elab 2403	. . . . . . . 8 |- (r e. {y | E.x e. B y = (F"x)} <-> E.x e. B r = (F"x))
117		imaeq2 4260	. . . . . . . . . 10 |- (x = u -> (F"x) = (F"u))
118	117	eqeq2d 1895	. . . . . . . . 9 |- (x = u -> (r = (F"x) <-> r = (F"u)))
119	118	cbvrexv 2281	. . . . . . . 8 |- (E.x e. B r = (F"x) <-> E.u e. B r = (F"u))
120	112, 116, 119	3bitri 194	. . . . . . 7 |- (r e. C <-> E.u e. B r = (F"u))
121	16	eleq2i 1961	. . . . . . . 8 |- (s e. C <-> s e. {y | E.x e. B y = (F"x)})
122		visset 2295	. . . . . . . . 9 |- s e. _V
123		eqeq1 1890	. . . . . . . . . 10 |- (y = s -> (y = (F"x) <-> s = (F"x)))
124	123	rexbidv 2124	. . . . . . . . 9 |- (y = s -> (E.x e. B y = (F"x) <-> E.x e. B s = (F"x)))
125	122, 124	elab 2403	. . . . . . . 8 |- (s e. {y | E.x e. B y = (F"x)} <-> E.x e. B s = (F"x))
126		imaeq2 4260	. . . . . . . . . 10 |- (x = v -> (F"x) = (F"v))
127	126	eqeq2d 1895	. . . . . . . . 9 |- (x = v -> (s = (F"x) <-> s = (F"v)))
128	127	cbvrexv 2281	. . . . . . . 8 |- (E.x e. B s = (F"x) <-> E.v e. B s = (F"v))
129	121, 125, 128	3bitri 194	. . . . . . 7 |- (s e. C <-> E.v e. B s = (F"v))
130	120, 129	anbi12i 540	. . . . . 6 |- ((r e. C /\ s e. C) <-> (E.u e. B r = (F"u) /\ E.v e. B s = (F"v)))
131		reeanv 2249	. . . . . 6 |- (E.u e. B E.v e. B (r = (F"u) /\ s = (F"v)) <-> (E.u e. B r = (F"u) /\ E.v e. B s = (F"v)))
132	130, 131	bitr4i 193	. . . . 5 |- ((r e. C /\ s e. C) <-> E.u e. B E.v e. B (r = (F"u) /\ s = (F"v)))
133	111, 132	syl5ib 223	. . . 4 |- ((B e. fBas /\ F Fn X) -> ((r e. C /\ s e. C) -> E.t e. C t C_ (r i^i s)))
134	133	r19.21aivv 2183	. . 3 |- ((B e. fBas /\ F Fn X) -> A.r e. C A.s e. C E.t e. C t C_ (r i^i s))
135	32, 73, 134	3jca 1050	. 2 |- ((B e. fBas /\ F Fn X) -> (C =/= (/) /\ (/) e/ C /\ A.r e. C A.s e. C E.t e. C t C_ (r i^i s)))
136		abrexexg 4837	. . . . 5 |- (B e. fBas -> {y | E.x e. B y = (F"x)} e. _V)
137	136, 16	syl5eqel 1975	. . . 4 |- (B e. fBas -> C e. _V)
138		isfbas2 10263	. . . 4 |- (C e. _V -> (C e. fBas <-> (C =/= (/) /\ (/) e/ C /\ A.r e. C A.s e. C E.t e. C t C_ (r i^i s))))
139	137, 138	syl 12	. . 3 |- (B e. fBas -> (C e. fBas <-> (C =/= (/) /\ (/) e/ C /\ A.r e. C A.s e. C E.t e. C t C_ (r i^i s))))
140	139	adantr 425	. 2 |- ((B e. fBas /\ F Fn X) -> (C e. fBas <-> (C =/= (/) /\ (/) e/ C /\ A.r e. C A.s e. C E.t e. C t C_ (r i^i s))))
141	135, 140	mpbird 213	1 |- ((B e. fBas /\ F Fn X) -> C e. fBas)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017   e/ wnel 2018  A.wral 2105  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875  U.cuni 3177  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  fBascfbas 10257

This theorem is referenced by:  filmapss 10309  fmf 10310  fmbas 10311  elfilmap 10312  cnpfillim 15589  fcluscnplem 15617

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-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-nel 2020  df-ral 2109  df-rex 2110  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-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-fbas 10259

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