Theorem filmapss 10309

Description: A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.)

Hypotheses

Ref	Expression
filmapss.1	|- Y = U.B
filmapss.2	|- Z = U.C

Assertion

Ref	Expression
filmapss	|- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ((X FilMap B)` F) C_ ((X FilMap C)` F))

Proof of Theorem filmapss

Step	Hyp	Ref	Expression
1		filmapss.1	. . . . . 6 |- Y = U.B
2		eqid 1884	. . . . . 6 |- {x | E.y e. B x = (F"y)} = {x | E.y e. B x = (F"y)}
3	1, 2	filrn 10293	. . . . 5 |- ((B e. fBas /\ F Fn Y) -> {x | E.y e. B x = (F"y)} e. fBas)
4		simp2 877	. . . . 5 |- ((X e. A /\ B e. fBas /\ C e. fBas) -> B e. fBas)
5		ffn 4562	. . . . . 6 |- (F:Y-->X -> F Fn Y)
6	5	3ad2ant1 897	. . . . 5 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> F Fn Y)
7	3, 4, 6	syl2an 503	. . . 4 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> {x | E.y e. B x = (F"y)} e. fBas)
8		sseq1 2637	. . . . . . . . . . . 12 |- (z = (F"y) -> (z C_ X <-> (F"y) C_ X))
9		imassrn 4278	. . . . . . . . . . . . . 14 |- (F"y) C_ ran F
10	9	a1i 8	. . . . . . . . . . . . 13 |- (F:Y-->X -> (F"y) C_ ran F)
11		frn 4569	. . . . . . . . . . . . 13 |- (F:Y-->X -> ran F C_ X)
12	10, 11	sstrd 2627	. . . . . . . . . . . 12 |- (F:Y-->X -> (F"y) C_ X)
13	8, 12	syl5cbir 228	. . . . . . . . . . 11 |- (F:Y-->X -> (z = (F"y) -> z C_ X))
14	13	a1d 15	. . . . . . . . . 10 |- (F:Y-->X -> (y e. B -> (z = (F"y) -> z C_ X)))
15	14	r19.23adv 2215	. . . . . . . . 9 |- (F:Y-->X -> (E.y e. B z = (F"y) -> z C_ X))
16		visset 2295	. . . . . . . . . 10 |- z e. _V
17		eqeq1 1890	. . . . . . . . . . 11 |- (x = z -> (x = (F"y) <-> z = (F"y)))
18	17	rexbidv 2124	. . . . . . . . . 10 |- (x = z -> (E.y e. B x = (F"y) <-> E.y e. B z = (F"y)))
19	16, 18	elab 2403	. . . . . . . . 9 |- (z e. {x | E.y e. B x = (F"y)} <-> E.y e. B z = (F"y))
20	15, 19	syl5ib 223	. . . . . . . 8 |- (F:Y-->X -> (z e. {x | E.y e. B x = (F"y)} -> z C_ X))
21	20	r19.21aiv 2175	. . . . . . 7 |- (F:Y-->X -> A.z e. {x | E.y e. B x = (F"y)}z C_ X)
22		unissb 3208	. . . . . . 7 |- (U.{x | E.y e. B x = (F"y)} C_ X <-> A.z e. {x | E.y e. B x = (F"y)}z C_ X)
23	21, 22	sylibr 217	. . . . . 6 |- (F:Y-->X -> U.{x | E.y e. B x = (F"y)} C_ X)
24	23	3ad2ant1 897	. . . . 5 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> U.{x | E.y e. B x = (F"y)} C_ X)
25	24	adantl 424	. . . 4 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> U.{x | E.y e. B x = (F"y)} C_ X)
26		eqid 1884	. . . . 5 |- U.{x | E.y e. B x = (F"y)} = U.{x | E.y e. B x = (F"y)}
27	26	extbas1 10291	. . . 4 |- (({x | E.y e. B x = (F"y)} e. fBas /\ U.{x | E.y e. B x = (F"y)} C_ X) -> ({x | E.y e. B x = (F"y)} u. {X}) e. fBas)
28	7, 25, 27	syl11anc 524	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ({x | E.y e. B x = (F"y)} u. {X}) e. fBas)
29		filmapss.2	. . . . . 6 |- Z = U.C
30		eqid 1884	. . . . . 6 |- {x | E.y e. C x = (F"y)} = {x | E.y e. C x = (F"y)}
31	29, 30	filrn 10293	. . . . 5 |- ((C e. fBas /\ F Fn Z) -> {x | E.y e. C x = (F"y)} e. fBas)
32		simp3 878	. . . . 5 |- ((X e. A /\ B e. fBas /\ C e. fBas) -> C e. fBas)
33		fneq2 4504	. . . . . . . 8 |- (Y = Z -> (F Fn Y <-> F Fn Z))
34	33	biimpac 462	. . . . . . 7 |- ((F Fn Y /\ Y = Z) -> F Fn Z)
35	34, 5	sylan 497	. . . . . 6 |- ((F:Y-->X /\ Y = Z) -> F Fn Z)
36	35	3adant3 896	. . . . 5 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> F Fn Z)
37	31, 32, 36	syl2an 503	. . . 4 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> {x | E.y e. C x = (F"y)} e. fBas)
38	13	a1d 15	. . . . . . . . . 10 |- (F:Y-->X -> (y e. C -> (z = (F"y) -> z C_ X)))
39	38	r19.23adv 2215	. . . . . . . . 9 |- (F:Y-->X -> (E.y e. C z = (F"y) -> z C_ X))
40	17	rexbidv 2124	. . . . . . . . . 10 |- (x = z -> (E.y e. C x = (F"y) <-> E.y e. C z = (F"y)))
41	16, 40	elab 2403	. . . . . . . . 9 |- (z e. {x | E.y e. C x = (F"y)} <-> E.y e. C z = (F"y))
42	39, 41	syl5ib 223	. . . . . . . 8 |- (F:Y-->X -> (z e. {x | E.y e. C x = (F"y)} -> z C_ X))
43	42	r19.21aiv 2175	. . . . . . 7 |- (F:Y-->X -> A.z e. {x | E.y e. C x = (F"y)}z C_ X)
44		unissb 3208	. . . . . . 7 |- (U.{x | E.y e. C x = (F"y)} C_ X <-> A.z e. {x | E.y e. C x = (F"y)}z C_ X)
45	43, 44	sylibr 217	. . . . . 6 |- (F:Y-->X -> U.{x | E.y e. C x = (F"y)} C_ X)
46	45	3ad2ant1 897	. . . . 5 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> U.{x | E.y e. C x = (F"y)} C_ X)
47	46	adantl 424	. . . 4 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> U.{x | E.y e. C x = (F"y)} C_ X)
48		eqid 1884	. . . . 5 |- U.{x | E.y e. C x = (F"y)} = U.{x | E.y e. C x = (F"y)}
49	48	extbas1 10291	. . . 4 |- (({x | E.y e. C x = (F"y)} e. fBas /\ U.{x | E.y e. C x = (F"y)} C_ X) -> ({x | E.y e. C x = (F"y)} u. {X}) e. fBas)
50	37, 47, 49	syl11anc 524	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ({x | E.y e. C x = (F"y)} u. {X}) e. fBas)
51		ssrexv 2673	. . . . . . . 8 |- (B C_ C -> (E.y e. B x = (F"y) -> E.y e. C x = (F"y)))
52	51	3ad2ant3 899	. . . . . . 7 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> (E.y e. B x = (F"y) -> E.y e. C x = (F"y)))
53	52	adantl 424	. . . . . 6 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> (E.y e. B x = (F"y) -> E.y e. C x = (F"y)))
54	53	19.21aiv 1664	. . . . 5 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> A.x(E.y e. B x = (F"y) -> E.y e. C x = (F"y)))
55		ss2ab 2675	. . . . 5 |- ({x | E.y e. B x = (F"y)} C_ {x | E.y e. C x = (F"y)} <-> A.x(E.y e. B x = (F"y) -> E.y e. C x = (F"y)))
56	54, 55	sylibr 217	. . . 4 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> {x | E.y e. B x = (F"y)} C_ {x | E.y e. C x = (F"y)})
57		unss1 2773	. . . 4 |- ({x | E.y e. B x = (F"y)} C_ {x | E.y e. C x = (F"y)} -> ({x | E.y e. B x = (F"y)} u. {X}) C_ ({x | E.y e. C x = (F"y)} u. {X}))
58	56, 57	syl 12	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ({x | E.y e. B x = (F"y)} u. {X}) C_ ({x | E.y e. C x = (F"y)} u. {X}))
59		fgss 10287	. . 3 |- ((({x | E.y e. B x = (F"y)} u. {X}) e. fBas /\ ({x | E.y e. C x = (F"y)} u. {X}) e. fBas /\ ({x | E.y e. B x = (F"y)} u. {X}) C_ ({x | E.y e. C x = (F"y)} u. {X})) -> (filGen` ({x | E.y e. B x = (F"y)} u. {X})) C_ (filGen` ({x | E.y e. C x = (F"y)} u. {X})))
60	28, 50, 58, 59	syl111anc 1100	. 2 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> (filGen` ({x | E.y e. B x = (F"y)} u. {X})) C_ (filGen` ({x | E.y e. C x = (F"y)} u. {X})))
61	1	isfilmap 10308	. . . . 5 |- ((X e. A /\ B e. fBas /\ F:Y-->X) -> ((X FilMap B)` F) = (filGen` ({x | E.y e. B x = (F"y)} u. {X})))
62	61	3expa 1067	. . . 4 |- (((X e. A /\ B e. fBas) /\ F:Y-->X) -> ((X FilMap B)` F) = (filGen` ({x | E.y e. B x = (F"y)} u. {X})))
63	62	3ad2antr1 1041	. . 3 |- (((X e. A /\ B e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ((X FilMap B)` F) = (filGen` ({x | E.y e. B x = (F"y)} u. {X})))
64	63	3adantl3 1034	. 2 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ((X FilMap B)` F) = (filGen` ({x | E.y e. B x = (F"y)} u. {X})))
65		simp1 876	. . . 4 |- ((X e. A /\ B e. fBas /\ C e. fBas) -> X e. A)
66	65	adantr 425	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> X e. A)
67	32	adantr 425	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> C e. fBas)
68		feq2 4552	. . . . . 6 |- (Y = Z -> (F:Y-->X <-> F:Z-->X))
69	68	biimpac 462	. . . . 5 |- ((F:Y-->X /\ Y = Z) -> F:Z-->X)
70	69	3adant3 896	. . . 4 |- ((F:Y-->X /\ Y = Z /\ B C_ C) -> F:Z-->X)
71	70	adantl 424	. . 3 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> F:Z-->X)
72	29	isfilmap 10308	. . 3 |- ((X e. A /\ C e. fBas /\ F:Z-->X) -> ((X FilMap C)` F) = (filGen` ({x | E.y e. C x = (F"y)} u. {X})))
73	66, 67, 71, 72	syl111anc 1100	. 2 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ((X FilMap C)` F) = (filGen` ({x | E.y e. C x = (F"y)} u. {X})))
74	60, 64, 73	3sstr4d 2660	1 |- (((X e. A /\ B e. fBas /\ C e. fBas) /\ (F:Y-->X /\ Y = Z /\ B C_ C)) -> ((X FilMap B)` F) C_ ((X FilMap C)` F))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106   u. cun 2591   C_ wss 2593  {csn 3044  U.cuni 3177  ran crn 3987  "cima 3989   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  fBascfbas 10257  filGencfg 10258   FilMap cfilmap 10304

This theorem is referenced by:  cnpfillim4 14947

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-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-fbas 10259  df-fg 10260  df-filmap 10306

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